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\hypersetup{pdftitle=Logiweb codex of sup2}

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\begin {document}\raggedright

\makebox [0mm][l]{\makebox [\linewidth ][r]{\includegraphics [0mm,18.5mm][16.5mm,19mm]{/home/disk07/grue/logiweb/20060417/html/logiweb/doc/logiweb.eps}}}%
{\bf {\Large Logiweb codex of $\rm sup2$}}

\vspace {12.5mm}

\href {../index.html}{Up}
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/browser/codex/index.html}{Help}


\begin {flushleft}\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:0}{$\rm sup2$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:1}{$\rm LeqTotality(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:2}{$\rm PositiveToLeft(Eq)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:3}{$\rm ExpZero(Exact) $},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:4}{$\rm SameExp(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:5}{$\rm SameExp(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:6}{$\rm SameExp$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:7}{$\rm (1/2)(x+y)-x=(1/2)(y-x)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:8}{$\rm y-(1/2)(x+y)=(1/2)(y-x)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:9}{$\rm BSzero(Exact)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:10}{$\rm SameBS(2)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:11}{$\rm SameBS(2)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:12}{$\rm SameBS(2)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:13}{$\rm NegativeToLeft(Less)(1 term)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:14}{$\rm BS(+1)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:15}{$\rm BSbound(Exact)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:16}{$\rm BSbound(Exact)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:17}{$\rm BSbound(Exact)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:18}{$\rm BSbound$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:19}{$\rm UStelescope(Zero)(Exact)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:20}{$\rm SameTelescope(2)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:21}{$\rm SameTelescope(2)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:22}{$\rm SameTelescope(2)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:23}{$\rm Exp(+1)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:24}{$\rm PositiveBase(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:25}{$\rm PositiveBase(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:26}{$\rm PositiveBase$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:27}{$\rm TelescopeNumerical(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:28}{$\rm TelescopeNumerical(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:29}{$\rm TelescopeNumerical$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:30}{$\rm (+1)IsPositive(N)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:31}{$\rm DistributionOut(Minus)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:32}{$\rm PositiveToRight(Eq)(1 term)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:33}{$\rm SameSeries(NumDiff)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:34}{$\rm ToNegatedDoubleImply$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:35}{$\rm AddNegatedAll$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:36}{$\rm (A)to(~E~)(Imply)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:37}{$\rm (E)to(~A~)(Imply)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:38}{$\rm (E~)to(~A)(Imply)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:39}{$\rm ToNegatedAEA $},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:40}{$\rm Three2threeTerms(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:41}{$\rm LessNeq(F)(Helper)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:42}{$\rm LessNeq(F)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:43}{$\rm LessNeq(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:44}{$\rm x=x+(y-y)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:45}{$\rm x=x+y-y(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:46}{$\rm SubtractEquations(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:47}{$\rm NeqAddition(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:48}{$\rm PositiveToRight(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:49}{$\rm PositiveToRight(Less)(1 term)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:50}{$\rm LeqNeqLess(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:51}{$\rm SubLeqLeft(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:52}{$\rm ToLeq(Advanced)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:53}{$\rm LeqLessTransitivity(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:54}{$\rm NegativeToLeft(Eq)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:55}{$\rm NegativeToRight(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:56}{$\rm !!==Symmetry$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:57}{$\rm SwitchTerms(x<=y-z)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:58}{$\rm Plus0Left(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:59}{$\rm PositiveToRight(Eq)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:60}{$\rm EqAdditionLeft(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:61}{$\rm Three2twoTerms(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:62}{$\rm To!!==$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:63}{$\rm PositiveToRight(Less)(1 term)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:64}{$\rm (A~)to(~E)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:65}{$\rm NegativeToRight(Eq)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:66}{$\rm NegativeToRight(Eq)(1 term)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:67}{$\rm DoubleMinus(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:68}{$\rm UniqueNegative(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:69}{$\rm SubtractEquationsLeft(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:70}{$\rm EqNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:71}{$\rm NeqNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:72}{$\rm -0=0(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:73}{$\rm NegativeNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:74}{$\rm FromLeqGeq(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:75}{$\rm 0<=|x|(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:76}{$\rm PositiveNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:77}{$\rm AddEquations(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:78}{$\rm Times(-1)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:79}{$\rm Times(-1)Left(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:80}{$\rm -x-y=-(x+y)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:81}{$\rm LessTotality(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:82}{$\rm SameNumerical(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:83}{$\rm MinusNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:84}{$\rm PositiveNumerical(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:85}{$\rm SignNumerical(+)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:86}{$\rm NonnegativeNumerical(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:87}{$\rm NegativeNumerical(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:88}{$\rm LeqNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:89}{$\rm LessNegated(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:90}{$\rm SubLeqRight(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:91}{$\rm FromLess(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:92}{$\rm DistributionOut(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:93}{$\rm x*0+x=x(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:94}{$\rm x*0=0(R)(fff)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:95}{$\rm SignNumerical(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:96}{$\rm NumericalDifference(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:97}{$\rm x<=|x|(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:98}{$\rm USlimitIsUpperBound(Helper)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:99}{$\rm USlimitIsUpperBound$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:100}{$\rm (-1)*(-1)+(-1)*1=0(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:101}{$\rm (-1)*(-1)=1(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:102}{$\rm 0<1Helper(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:103}{$\rm 0<1(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:104}{$\rm ExpZero(Exact)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:105}{$\rm PositiveBase(R)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:106}{$\rm Three2twoFactors(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:107}{$\rm x=x*y*(1/y)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:108}{$\rm NeqMultiplication(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:109}{$\rm LessTransitivity(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:110}{$\rm 0<2(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:111}{$\rm SameExp(R)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:112}{$\rm SameExp(R)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:113}{$\rm SameExp(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:114}{$\rm SubNeqLeft(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:115}{$\rm SubNeqRight(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:116}{$\rm NonzeroFactors(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:117}{$\rm NonnegativeFactors(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:118}{$\rm PositiveFactors(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:119}{$\rm LessDivision(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:120}{$\rm 0<1/2(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:121}{$\rm PositiveToRight(Eq)(1 term)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:122}{$\rm Exp(+1)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:123}{$\rm PositiveBase(R)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:124}{$\rm PositiveBase(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:125}{$\rm -x*y=-(x*y)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:126}{$\rm PositiveToLeft(Eq)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:127}{$\rm Times1Left(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:128}{$\rm x+x=2*x(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:129}{$\rm (1/2)x+(1/2)x=x(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:130}{$\rm DistributionOut(Minus)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:131}{$\rm (1/2)(x+y)-x=(1/2)(y-x)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:132}{$\rm IntervalSize(R)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:133}{$\rm LessMultiplicationLeft(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:134}{$\rm NegativeToLeft(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:135}{$\rm NegativeToLeft(Less)(1 term)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:136}{$\rm y-(1/2)(x+y)=(1/2)(y-x)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:137}{$\rm IntervalSize(R)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:138}{$\rm IntervalSize(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:139}{$\rm XSlessUS(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:140}{$\rm USdecreasing(+1)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:141}{$\rm ExpUnbounded(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:142}{$\rm ExpUnbounded(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:143}{$\rm ExpUnbounded$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:144}{$\rm 1<=x+1(N)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:145}{$\rm ExpNonzero(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:146}{$\rm ExpNonzero(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:147}{$\rm ExpNonzero$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:148}{$\rm ExpNonzero(2)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:149}{$\rm HalfBase(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:150}{$\rm HalfBase(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:151}{$\rm MultiplyEquations(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:152}{$\rm NonreciprocalToRight(Eq)(1 term)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:153}{$\rm PositiveNonzero(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:154}{$\rm NonzeroProduct(2)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:155}{$\rm HalfBase$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:156}{$\rm Three2threeFactors(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:157}{$\rm x*y=zBackwards(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:158}{$\rm PositiveInverted(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:159}{$\rm ReciprocalToRight(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:160}{$\rm ReciprocalToRight(Less)(1 term)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:161}{$\rm NonreciprocalToLeft(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:162}{$\rm 1<x*y(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:163}{$\rm SwitchFactors(1/x<y)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:164}{$\rm SmallHalving$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:165}{$\rm IntervalSize(anyPositive)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:166}{$\rm USdecreasing(+n)(Base)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:167}{$\rm USdecreasing(+n)(Indu)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:168}{$\rm USdecreasing(+n)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:169}{$\rm USdecreasing$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:170}{$\rm LeqAdditionLeft(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:171}{$\rm ToNotLess(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:172}{$\rm LimitOfUSIsLeq$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:173}{$\rm SubtractEquations(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:174}{$\rm SubtractEquationsLeft(Less)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:175}{$\rm LessNegated(Negative)(R)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:176}{$\rm FromNegatedAnd(Imply)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:177}{$\rm RemoveDoubleNeg(Consequent)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:178}{$\rm FromNotUpperBound$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:179}{$\rm LeqNUB$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:180}{$\rm USlimitIsLeastUpperBound(Helper)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:181}{$\rm USlimitIsLeastUpperBound$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:182}{$\rm ExistMP3$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:183}{$\rm GreaterPositive(N)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:184}{$\rm ysFClose(Helper)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:185}{$\rm ysFClose$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:186}{$\rm ysFCauchy(Helper)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:187}{$\rm ysFCauchy$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:188}{$\rm from<<==$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:189}{$\rm to<<==$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:190}{$\rm NonnegativeNumerical(F)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:191}{$\rm NegativeNumerical(F)$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:192}{$\rm 
\mbox{tester1}$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:193}{$\rm 
\mbox{tester2}$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:194}{$\rm 
\mbox{tester3}$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:195}{$\rm 
\mbox{tester4}$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:196}{$\rm 
\mbox{tester5}$},
\hyperlink {url:1808490797518007916660448878599179667361805706605856764796691709371934721:197}{$\rm 
\mbox{tester6}$},
\end {flushleft}

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:0}{}\section* {$\rm sup2$}

$\rm
[sup2
\stackrel{
\mathrm{prio}
}{\rightarrow}
\newline \mathbf {Preassociative} \newline 
[sup2
] , \linebreak [0] 
[
\mbox{base}
] , \linebreak [0] 
[
\mbox{bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\mbox{big bracket $
\ast $ end bracket}
] , \linebreak [0] 
[
\ \$
\ast \$\linebreak[0]\ 
] , \linebreak [0] 
[
\mathbf{flush\ left\ }[ 
\ast 
]
] , \linebreak [0] 
[
\mathsf{x}
] , \linebreak [0] 
[
\mathsf{y}
] , \linebreak [0] 
[
\mathsf{z}
] , \linebreak [0] 
[
[
\ast 
\bowtie
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel{
\ast 
}{\rightarrow}
\ast 
]
] , \linebreak [0] 
[
\mathrm{pyk}
] , \linebreak [0] 
[
\mathrm{tex}
] , \linebreak [0] 
[
\mathrm{name}
] , \linebreak [0] 
[
\mathrm{prio}
] , \linebreak [0] 
[
\ast 
] , \linebreak [0] 
[
\mathsf {T}
] , \linebreak [0] 
[
\mathrm {if}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {
\ast 
}{\Rightarrow }
\ast 
]
] , \linebreak [0] 
[
\mathrm {val}
] , \linebreak [0] 
[
\mathrm {claim}
] , \linebreak [0] 
[
\bot 
] , \linebreak [0] 
[
f(
\ast 
)
] , \linebreak [0] 
[
(
\ast 
){}^{I} 
] , \linebreak [0] 
[
\mathsf {F}
] , \linebreak [0] 
[
\underline {0}
] , \linebreak [0] 
[
\underline {1}
] , \linebreak [0] 
[
\underline {2}
] , \linebreak [0] 
[
\underline {3}
] , \linebreak [0] 
[
\underline {4}
] , \linebreak [0] 
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\underline {5}
] , \linebreak [0] 
[
\underline {6}
] , \linebreak [0] 
[
\underline {7}
] , \linebreak [0] 
[
\underline {8}
] , \linebreak [0] 
[
\underline {9}
] , \linebreak [0] 
[
0
] , \linebreak [0] 
[
1
] , \linebreak [0] 
[
2
] , \linebreak [0] 
[
3
] , \linebreak [0] 
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] , \linebreak [0] 
[
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] , \linebreak [0] 
[
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] , \linebreak [0] 
[
7
] , \linebreak [0] 
[
8
] , \linebreak [0] 
[
9
] , \linebreak [0] 
[
\mathsf {a}
] , \linebreak [0] 
[
\mathsf {b}
] , \linebreak [0] 
[
\mathsf {c}
] , \linebreak [0] 
[
\mathsf {d}
] , \linebreak [0] 
[
\mathsf {e}
] , \linebreak [0] 
[
\mathsf {f}
] , \linebreak [0] 
[
\mathsf {g}
] , \linebreak [0] 
[
\mathsf {h}
] , \linebreak [0] 
[
\mathsf {i}
] , \linebreak [0] 
[
\mathsf {j}
] , \linebreak [0] 
[
\mathsf {k}
] , \linebreak [0] 
[
\mathsf {l}
] , \linebreak [0] 
[
\mathsf {m}
] , \linebreak [0] 
[
\mathsf {n}
] , \linebreak [0] 
[
\mathsf {o}
] , \linebreak [0] 
[
\mathsf {p}
] , \linebreak [0] 
[
\mathsf {q}
] , \linebreak [0] 
[
\mathsf {r}
] , \linebreak [0] 
[
\mathsf {s}
] , \linebreak [0] 
[
\mathsf {t}
] , \linebreak [0] 
[
\mathsf {u}
] , \linebreak [0] 
[
\mathsf {v}
] , \linebreak [0] 
[
\mathsf {w}
] , \linebreak [0] 
[
(
\ast 
)^M
] , \linebreak [0] 
[
\mathrm {If}(
\ast 
,\linebreak [0]
\ast 
,\linebreak [0]
\ast 
)
] , \linebreak [0] 
[\mathrm {array}\{
\ast 
\}
\ast 
\mathrm {end\ array}
] , \linebreak [0] 
[
l
] , \linebreak [0] 
[
c
] , \linebreak [0] 
[
r
] , \linebreak [0] 
[
\mathrm {empty}
] , \linebreak [0] 
[
\langle 
\ast 
\,{|}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
{\cal M}(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal U}}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}(
\ast 
)
] , \linebreak [0] 
[
{\cal U}^M(
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {apply}_1(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
identifier( 
\ast 
)
] , \linebreak [0] 
[
identifier_{1}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]remove( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]put( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
array\mbox{-}\linebreak[0]add( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
bit_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rack
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {vector}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {bibliography}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {dictionary}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {body}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {codex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {expansion}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {code}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {cache}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {diagnose}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pyk}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {tex}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {texname}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {value}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {message}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {macro}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {definition}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {unpack}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {claim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {priority}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {lambda}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {apply}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {true}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {if}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {quote}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {proclaim}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {define}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {introduce}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {hide}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {pre}\mbox {\tt \char34}
] , \linebreak [0] 
[
\mbox {\tt \char34}\mathrm {post}\mbox {\tt \char34}
] , \linebreak [0] 
[
{\cal E}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal E}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {lookup}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {abstract}(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\lceil 
\ast 
\rceil 
] , \linebreak [0] 
[
{\cal M}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}_2(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
{\cal M}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathrm {macro}
] , \linebreak [0] 
[
s_0
] , \linebreak [0] 
[
\mathbf {zip}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\mathbf {assoc}_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)^{\mathbf {p}}
] , \linebreak [0] 
[
\mbox{self}
] , \linebreak [0] 
[
[
\ast 
\mathrel {\ddot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\dot {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\mathrel {\acute {=}}
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {pyk}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {tex}}{=} 
\ast 
]
] , \linebreak [0] 
[
[
\ast 
\stackrel {\mathrm {name}}{=} 
\ast 
]
] , \linebreak [0] 
[
\mathbf{Priority\ table} [ 
\ast 
]
] , \linebreak [0] 
[
\tilde {{\cal M}}_1
] , \linebreak [0] 
[
\tilde {{\cal M}}_2(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_3(
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal M}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\tilde {{\cal Q}}^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
(
\ast 
)
] , \linebreak [0] 
[
\left( 
\ast 
\right) 
] , \linebreak [0] 
[
display(
\ast 
)
] , \linebreak [0] 
[
statement(
\ast 
)
] , \linebreak [0] 
[
[
\ast 
]^{\cdot}
] , \linebreak [0] 
[
[
\ast 
]^-
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{aspect}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\langle 
\ast 
\rangle 
] , \linebreak [0] 
[
\mathbf {tuple}_1( 
\ast 
) 
] , \linebreak [0] 
[
\mathbf {tuple}_2( 
\ast 
) 
] , \linebreak [0] 
[
let_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
let_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[
\ast 
\stackrel {claim}{=}
\ast 
]
] , \linebreak [0] 
[
\mbox{checker}
] , \linebreak [0] 
[
\mathbf{check}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}_3( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mathbf{check}^*_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\cdot} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{-} 
] , \linebreak [0] 
[
\relax [ 
\ast 
\relax ]^{\circ} 
] , \linebreak [0] 
[
msg
] , \linebreak [0] 
[
[
\ast 
\stackrel {msg}{=} 
\ast 
]
] , \linebreak [0] 
[
{<}stmt{>}
] , \linebreak [0] 
[
stmt
] , \linebreak [0] 
[
[
\ast 
\stackrel {stmt}{=}
\ast 
]
] , \linebreak [0] 
[
HeadNil'
] , \linebreak [0] 
[
HeadPair'
] , \linebreak [0] 
[
Transitivity'
] , \linebreak [0] 
[
{\makebox [0mm][l]{$\bot $}\,{\bot }}
] , \linebreak [0] 
[
Contra'
] , \linebreak [0] 
[
T'_{E}
] , \linebreak [0] 
[
L_{1}
] , \linebreak [0] 
[\underline{
\ast }
] , \linebreak [0] 
[{\cal A}
] , \linebreak [0] 
[{\cal B}
] , \linebreak [0] 
[{\cal C}
] , \linebreak [0] 
[{\cal D}
] , \linebreak [0] 
[{\cal E}
] , \linebreak [0] 
[{\cal F}
] , \linebreak [0] 
[{\cal G}
] , \linebreak [0] 
[{\cal H}
] , \linebreak [0] 
[{\cal I}
] , \linebreak [0] 
[{\cal J}
] , \linebreak [0] 
[{\cal K}
] , \linebreak [0] 
[{\cal L}
] , \linebreak [0] 
[{\cal M}
] , \linebreak [0] 
[{\cal N}
] , \linebreak [0] 
[{\cal O}
] , \linebreak [0] 
[{\cal P}
] , \linebreak [0] 
[{\cal Q}
] , \linebreak [0] 
[{\cal R}
] , \linebreak [0] 
[{\cal S}
] , \linebreak [0] 
[{\cal T}
] , \linebreak [0] 
[{\cal U}
] , \linebreak [0] 
[{\cal V}
] , \linebreak [0] 
[{\cal W}
] , \linebreak [0] 
[{\cal X}
] , \linebreak [0] 
[{\cal Y}
] , \linebreak [0] 
[{\cal Z}
] , \linebreak [0] 
[
\langle 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\langle ^ { \ast } 
\ast 
\,{\protect\vert}
\ast 
{:=}\,
\ast 
\rangle 
] , \linebreak [0] 
[
\emptyset 
] , \linebreak [0] 
[
Remainder
] , \linebreak [0] 
[
( 
\ast 
)^{\bf v}
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
error_{2}(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
proof( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
proof_{2}( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{I}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\rhd}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\rhd}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{E}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{E}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{+}(
\ast 
, 
\ast 
)
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[
{\cal S}_{1}^{+}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{-}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{-}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\ast}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{\ast}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\char64}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\char64}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\vdash}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\vdash}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\makebox [0mm][l]{\scriptsize $\vdash $}\,{\vdash }}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{i.e.}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{i.e.}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{\forall}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{\forall}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}^{;}(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{1}^{;}(
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal S}_{2}^{;}(
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal T}( 
\ast 
)
] , \linebreak [0] 
[
claims( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
claims_2( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{<}proof{>}
] , \linebreak [0] 
[
proof
] , \linebreak [0] 
[
[ \mathbf{Lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ \mathbf{Proof\ of\ }
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ lemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antilemma\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ rule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
[ 
\ast 
\mathbf{\ antirule\ } 
\ast 
\colon 
\ast 
]
] , \linebreak [0] 
[
\mbox{verifier}
] , \linebreak [0] 
[
{\cal V}_1( 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_3( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_4( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_5( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_6( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal V}_7( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Cut( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Head_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
Tail_{\oplus} ( 
\ast 
)
] , \linebreak [0] 
[
rule_1( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
rule( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
Rule\ tactic
] , \linebreak [0] 
[
Plus( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
[ \mathbf{Theory\ } 
\ast 
]
] , \linebreak [0] 
[
theory_2( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_3( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
theory_4( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
HeadNil''
] , \linebreak [0] 
[
HeadPair''
] , \linebreak [0] 
[
Transitivity''
] , \linebreak [0] 
[
Contra''
] , \linebreak [0] 
[
HeadNil
] , \linebreak [0] 
[
HeadPair
] , \linebreak [0] 
[
Transitivity
] , \linebreak [0] 
[
Contra
] , \linebreak [0] 
[
T_E
] , \linebreak [0] 
[
ragged\ right
] , \linebreak [0] 
[
ragged\ right\ expansion\ 
] , \linebreak [0] 
[
parm(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
parm^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
inst^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
occur^*(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify^*(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[
unify_2(
\ast 
=
\ast 
,
\ast 
)
] , \linebreak [0] 
[L_a
] , \linebreak [0] 
[L_b
] , \linebreak [0] 
[L_c
] , \linebreak [0] 
[L_d
] , \linebreak [0] 
[L_e
] , \linebreak [0] 
[L_f
] , \linebreak [0] 
[L_g
] , \linebreak [0] 
[L_h
] , \linebreak [0] 
[L_i
] , \linebreak [0] 
[L_j
] , \linebreak [0] 
[L_k
] , \linebreak [0] 
[L_l
] , \linebreak [0] 
[L_m
] , \linebreak [0] 
[L_n
] , \linebreak [0] 
[L_o
] , \linebreak [0] 
[L_p
] , \linebreak [0] 
[L_q
] , \linebreak [0] 
[L_r
] , \linebreak [0] 
[L_s
] , \linebreak [0] 
[L_t
] , \linebreak [0] 
[L_u
] , \linebreak [0] 
[L_v
] , \linebreak [0] 
[L_w
] , \linebreak [0] 
[L_x
] , \linebreak [0] 
[L_y
] , \linebreak [0] 
[L_z
] , \linebreak [0] 
[L_A
] , \linebreak [0] 
[L_B
] , \linebreak [0] 
[L_C
] , \linebreak [0] 
[L_D
] , \linebreak [0] 
[L_E
] , \linebreak [0] 
[L_F
] , \linebreak [0] 
[L_G
] , \linebreak [0] 
[L_H
] , \linebreak [0] 
[L_I
] , \linebreak [0] 
[L_J
] , \linebreak [0] 
[L_K
] , \linebreak [0] 
[L_L
] , \linebreak [0] 
[L_M
] , \linebreak [0] 
[L_N
] , \linebreak [0] 
[L_O
] , \linebreak [0] 
[L_P
] , \linebreak [0] 
[L_Q
] , \linebreak [0] 
[L_R
] , \linebreak [0] 
[L_S
] , \linebreak [0] 
[L_T
] , \linebreak [0] 
[L_U
] , \linebreak [0] 
[L_V
] , \linebreak [0] 
[L_W
] , \linebreak [0] 
[L_X
] , \linebreak [0] 
[L_Y
] , \linebreak [0] 
[L_Z
] , \linebreak [0] 
[L_?
] , \linebreak [0] 
[
Reflexivity
] , \linebreak [0] 
[
Reflexivity_1
] , \linebreak [0] 
[
Commutativity
] , \linebreak [0] 
[
Commutativity_1
] , \linebreak [0] 
[
{<}tactic{>}
] , \linebreak [0] 
[
tactic
] , \linebreak [0] 
[
[
\ast 
\stackrel {tactic}{=}
\ast 
]
] , \linebreak [0] 
[
{\cal P}( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
{\cal P}^*( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
p_0
] , \linebreak [0] 
[
conclude_1 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_2 ( 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_3 ( 
\ast 
, 
\ast 
, 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
conclude_4 ( 
\ast 
, 
\ast 
)
] , \linebreak [0] 
[
\mbox{check}
] , \linebreak [0] 
[
[
\ast 
\stackrel {\circ}{=}
\ast 
]
] , \linebreak [0] 
[
RootVisible(
\ast 
)
] , \linebreak [0] 
[
A
] , \linebreak [0] 
[
R
] , \linebreak [0] 
[
C
] , \linebreak [0] 
[
T
] , \linebreak [0] 
[
L
] , \linebreak [0] 
[
\{
\ast 
\}
] , \linebreak [0] 
[\overline{
\ast }
] , \linebreak [0] 
[
\mathit{a}
] , \linebreak [0] 
[
\mathit{b}
] , \linebreak [0] 
[
\mathit{c}
] , \linebreak [0] 
[
\mathit{d}
] , \linebreak [0] 
[
\mathit{e}
] , \linebreak [0] 
[
\mathit{f}
] , \linebreak [0] 
[
\mathit{g}
] , \linebreak [0] 
[
\mathit{h}
] , \linebreak [0] 
[
\mathit{i}
] , \linebreak [0] 
[
\mathit{j}
] , \linebreak [0] 
[
\mathit{k}
] , \linebreak [0] 
[
\mathit{l}
] , \linebreak [0] 
[
\mathit{m}
] , \linebreak [0] 
[
\mathit{n}
] , \linebreak [0] 
[
\mathit{o}
] , \linebreak [0] 
[
\mathit{p}
] , \linebreak [0] 
[
\mathit{q}
] , \linebreak [0] 
[
\mathit{r}
] , \linebreak [0] 
[
\mathit{s}
] , \linebreak [0] 
[
\mathit{t}
] , \linebreak [0] 
[
\mathit{u}
] , \linebreak [0] 
[
\mathit{v}
] , \linebreak [0] 
[
\mathit{w}
] , \linebreak [0] 
[
\mathit{x}
] , \linebreak [0] 
[
\mathit{y}
] , \linebreak [0] 
[
\mathit{z}
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:=} 
\ast 
\rangle 
] , \linebreak [0] 
[
Ded(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_0(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_2(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_7(
\ast 
)
] , \linebreak [0] 
[
Ded_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Ded_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
S
] , \linebreak [0] 
[
Neg
] , \linebreak [0] 
[
MP
] , \linebreak [0] 
[
Gen
] , \linebreak [0] 
[
Ded
] , \linebreak [0] 
[
S1
] , \linebreak [0] 
[
S2
] , \linebreak [0] 
[
S3
] , \linebreak [0] 
[
S4
] , \linebreak [0] 
[
S5
] , \linebreak [0] 
[
S6
] , \linebreak [0] 
[
S7
] , \linebreak [0] 
[
S8
] , \linebreak [0] 
[
S9
] , \linebreak [0] 
[
Repetition
] , \linebreak [0] 
[
A1'
] , \linebreak [0] 
[
A2'
] , \linebreak [0] 
[
A4'
] , \linebreak [0] 
[
A5'
] , \linebreak [0] 
[
Prop\ 3.2a
] , \linebreak [0] 
[
Prop\ 3.2b
] , \linebreak [0] 
[
Prop\ 3.2c
] , \linebreak [0] 
[
Prop\ 3.2d
] , \linebreak [0] 
[
Prop\ 3.2e_1
] , \linebreak [0] 
[
Prop\ 3.2e_2
] , \linebreak [0] 
[
Prop\ 3.2e
] , \linebreak [0] 
[
Prop\ 3.2f_1
] , \linebreak [0] 
[
Prop\ 3.2f_2
] , \linebreak [0] 
[
Prop\ 3.2f
] , \linebreak [0] 
[
Prop\ 3.2g_1
] , \linebreak [0] 
[
Prop\ 3.2g_2
] , \linebreak [0] 
[
Prop\ 3.2g
] , \linebreak [0] 
[
Prop\ 3.2h_1
] , \linebreak [0] 
[
Prop\ 3.2h_2
] , \linebreak [0] 
[
Prop\ 3.2h
] , \linebreak [0] 
[
Block_1(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Block_2(
\ast 
)
] , \linebreak [0] 
[kvanti
] , \linebreak [0] 
[UniqueMember
] , \linebreak [0] 
[UniqueMember(Type)
] , \linebreak [0] 
[SameSeries
] , \linebreak [0] 
[A4
] , \linebreak [0] 
[SameMember
] , \linebreak [0] 
[Qclosed(Addition)
] , \linebreak [0] 
[Qclosed(Multiplication)
] , \linebreak [0] 
[FromCartProd(1)
] , \linebreak [0] 
[
\mbox{1rule fromCartProd(2)}
] , \linebreak [0] 
[constantRationalSeries(
\ast 
)
] , \linebreak [0] 
[cartProd(
\ast 
)
] , \linebreak [0] 
[Power(
\ast 
)
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[binaryUnion(
\ast 
,
\ast 
)
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[SetOfRationalSeries
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[IsSubset(
\ast 
,
\ast 
)
] , \linebreak [0] 
[(p
\ast 
,
\ast 
)
] , \linebreak [0] 
[(s
\ast 
)
] , \linebreak [0] 
[(\cdots{})
] , \linebreak [0] 
[\texttt{Objekt-var}
] , \linebreak [0] 
[\texttt{Ex-var}
] , \linebreak [0] 
[\texttt{Ph-var}
] , \linebreak [0] 
[\texttt{V\ae{}rdi}
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[\texttt{Variabel}
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[Op(
\ast 
)
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[Op(
\ast 
,
\ast 
)
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[
\ast 
 \mathrel {\ddot {==}} 
\ast 
] , \linebreak [0] 
[ContainsEmpty(
\ast 
)
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[Nat(
\ast 
)
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[
Dedu(
\ast 
,
\ast 
)
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[
Dedu_0(
\ast 
,
\ast 
)
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[Dedu_{s}(
\ast 
,
\ast 
,
\ast 
)
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[
Dedu_1(
\ast 
,
\ast 
,
\ast 
)
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[
Dedu_2(
\ast 
,
\ast 
,
\ast 
)
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[
Dedu_3(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
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[
Dedu_4(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_4^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_5(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_6^*(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_7(
\ast 
)
] , \linebreak [0] 
[
Dedu_8(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
Dedu_8^*(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Ex_{1}
] , \linebreak [0] 
[Ex_{2}
] , \linebreak [0] 
[Ex3
] , \linebreak [0] 
[Ex_{10}
] , \linebreak [0] 
[Ex_{20}
] , \linebreak [0] 
[
\ast 
 _{Ex}
] , \linebreak [0] 
[
\ast 
 ^{Ex}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ex} 
] , \linebreak [0] 
[ph_{1}
] , \linebreak [0] 
[ph_{2}
] , \linebreak [0] 
[ph_{3}
] , \linebreak [0] 
[
\ast 
 _{Ph} 
] , \linebreak [0] 
[
\ast 
 ^{Ph}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^0 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Ph} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv} 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me}
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^1 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
] , \linebreak [0] 
[\langle 
\ast 
{\equiv}^* 
\ast 
| 
\ast 
{:==} 
\ast 
\rangle_{Me} 
] , \linebreak [0] 
[\mathsf {bs}
] , \linebreak [0] 
[ \mathsf {OBS}
] , \linebreak [0] 
[{\cal BS}
] , \linebreak [0] 
[\mathrm{\O}
] , \linebreak [0] 
[SystemQ
] , \linebreak [0] 
[MP
] , \linebreak [0] 
[Gen
] , \linebreak [0] 
[Repetition
] , \linebreak [0] 
[Neg
] , \linebreak [0] 
[Ded
] , \linebreak [0] 
[ExistIntro
] , \linebreak [0] 
[Extensionality
] , \linebreak [0] 
[\O{}def
] , \linebreak [0] 
[PairDef
] , \linebreak [0] 
[UnionDef
] , \linebreak [0] 
[PowerDef
] , \linebreak [0] 
[SeparationDef
] , \linebreak [0] 
[AddDoubleNeg
] , \linebreak [0] 
[RemoveDoubleNeg
] , \linebreak [0] 
[AndCommutativity
] , \linebreak [0] 
[AutoImply
] , \linebreak [0] 
[Contrapositive
] , \linebreak [0] 
[FirstConjunct
] , \linebreak [0] 
[SecondConjunct
] , \linebreak [0] 
[FromContradiction
] , \linebreak [0] 
[FromDisjuncts
] , \linebreak [0] 
[IffCommutativity
] , \linebreak [0] 
[IffFirst
] , \linebreak [0] 
[IffSecond
] , \linebreak [0] 
[ImplyTransitivity
] , \linebreak [0] 
[JoinConjuncts
] , \linebreak [0] 
[MP2
] , \linebreak [0] 
[MP3
] , \linebreak [0] 
[MP4
] , \linebreak [0] 
[MP5
] , \linebreak [0] 
[MT
] , \linebreak [0] 
[NegativeMT
] , \linebreak [0] 
[Technicality
] , \linebreak [0] 
[Weakening
] , \linebreak [0] 
[WeakenOr1
] , \linebreak [0] 
[WeakenOr2
] , \linebreak [0] 
[Formula2Pair
] , \linebreak [0] 
[Pair2Formula
] , \linebreak [0] 
[Formula2Union
] , \linebreak [0] 
[Union2Formula
] , \linebreak [0] 
[Formula2Sep
] , \linebreak [0] 
[Sep2Formula
] , \linebreak [0] 
[Formula2Power
] , \linebreak [0] 
[SubsetInPower
] , \linebreak [0] 
[HelperPowerIsSub
] , \linebreak [0] 
[PowerIsSub
] , \linebreak [0] 
[(Switch)HelperPowerIsSub
] , \linebreak [0] 
[(Switch)PowerIsSub
] , \linebreak [0] 
[ToSetEquality
] , \linebreak [0] 
[HelperToSetEquality(t)
] , \linebreak [0] 
[ToSetEquality(t)
] , \linebreak [0] 
[HelperFromSetEquality
] , \linebreak [0] 
[FromSetEquality
] , \linebreak [0] 
[HelperReflexivity
] , \linebreak [0] 
[Reflexivity
] , \linebreak [0] 
[HelperSymmetry
] , \linebreak [0] 
[Symmetry
] , \linebreak [0] 
[HelperTransitivity
] , \linebreak [0] 
[Transitivity
] , \linebreak [0] 
[ERisReflexive
] , \linebreak [0] 
[ERisSymmetric
] , \linebreak [0] 
[ERisTransitive
] , \linebreak [0] 
[\O{}isSubset
] , \linebreak [0] 
[HelperMemberNot\O{}
] , \linebreak [0] 
[MemberNot\O{}
] , \linebreak [0] 
[HelperUnique\O{}
] , \linebreak [0] 
[Unique\O{}
] , \linebreak [0] 
[==\!{}Reflexivity
] , \linebreak [0] 
[==\!{}Symmetry
] , \linebreak [0] 
[Helper\!{}==\!{}Transitivity
] , \linebreak [0] 
[\!{}==\!{}Transitivity
] , \linebreak [0] 
[HelperTransferNotEq
] , \linebreak [0] 
[TransferNotEq
] , \linebreak [0] 
[HelperPairSubset
] , \linebreak [0] 
[Helper(2)PairSubset
] , \linebreak [0] 
[PairSubset
] , \linebreak [0] 
[SamePair
] , \linebreak [0] 
[SameSingleton
] , \linebreak [0] 
[UnionSubset
] , \linebreak [0] 
[SameUnion
] , \linebreak [0] 
[SeparationSubset
] , \linebreak [0] 
[SameSeparation
] , \linebreak [0] 
[SameBinaryUnion
] , \linebreak [0] 
[IntersectionSubset
] , \linebreak [0] 
[SameIntersection
] , \linebreak [0] 
[AutoMember
] , \linebreak [0] 
[HelperEqSysNot\O{}
] , \linebreak [0] 
[EqSysNot\O{}
] , \linebreak [0] 
[HelperEqSubset
] , \linebreak [0] 
[EqSubset
] , \linebreak [0] 
[HelperEqNecessary
] , \linebreak [0] 
[EqNecessary
] , \linebreak [0] 
[HelperNoneEqNecessary
] , \linebreak [0] 
[Helper(2)NoneEqNecessary
] , \linebreak [0] 
[NoneEqNecessary
] , \linebreak [0] 
[EqClassIsSubset
] , \linebreak [0] 
[EqClassesAreDisjoint
] , \linebreak [0] 
[AllDisjoint
] , \linebreak [0] 
[AllDisjointImply
] , \linebreak [0] 
[BSsubset
] , \linebreak [0] 
[Union(BS/R)subset
] , \linebreak [0] 
[UnionIdentity
] , \linebreak [0] 
[EqSysIsPartition
] , \linebreak [0] 
[(x1)
] , \linebreak [0] 
[(x2)
] , \linebreak [0] 
[(y1)
] , \linebreak [0] 
[(y2)
] , \linebreak [0] 
[(v1)
] , \linebreak [0] 
[(v2)
] , \linebreak [0] 
[(v3)
] , \linebreak [0] 
[(v4)
] , \linebreak [0] 
[(v2n)
] , \linebreak [0] 
[(m1)
] , \linebreak [0] 
[(m2)
] , \linebreak [0] 
[(n1)
] , \linebreak [0] 
[(n2)
] , \linebreak [0] 
[(n3)
] , \linebreak [0] 
[(\epsilon)
] , \linebreak [0] 
[(\epsilon)_{1}
] , \linebreak [0] 
[(\epsilon 2)
] , \linebreak [0] 
[(fep)
] , \linebreak [0] 
[(fx)
] , \linebreak [0] 
[(fy)
] , \linebreak [0] 
[(fz)
] , \linebreak [0] 
[(fu)
] , \linebreak [0] 
[(fv)
] , \linebreak [0] 
[(fw)
] , \linebreak [0] 
[(rx)
] , \linebreak [0] 
[(ry)
] , \linebreak [0] 
[(rz)
] , \linebreak [0] 
[(ru)
] , \linebreak [0] 
[(sx)
] , \linebreak [0] 
[(sx1)
] , \linebreak [0] 
[(sy)
] , \linebreak [0] 
[(sy1)
] , \linebreak [0] 
[(sz)
] , \linebreak [0] 
[(sz1)
] , \linebreak [0] 
[(su)
] , \linebreak [0] 
[(su1)
] , \linebreak [0] 
[(fxs)
] , \linebreak [0] 
[(fys)
] , \linebreak [0] 
[(crs1)
] , \linebreak [0] 
[(f1)
] , \linebreak [0] 
[(f2)
] , \linebreak [0] 
[(f3)
] , \linebreak [0] 
[(f4)
] , \linebreak [0] 
[(op1)
] , \linebreak [0] 
[(op2)
] , \linebreak [0] 
[(r1)
] , \linebreak [0] 
[(s1)
] , \linebreak [0] 
[(s2)
] , \linebreak [0] 
[X_{1}
] , \linebreak [0] 
[X_{2}
] , \linebreak [0] 
[Y_{1}
] , \linebreak [0] 
[Y_{2}
] , \linebreak [0] 
[V_{1}
] , \linebreak [0] 
[V_{2}
] , \linebreak [0] 
[V_{3}
] , \linebreak [0] 
[V_{4}
] , \linebreak [0] 
[V_{2n}
] , \linebreak [0] 
[M_{1}
] , \linebreak [0] 
[M_{2}
] , \linebreak [0] 
[N_{1} 
] , \linebreak [0] 
[N_{2} 
] , \linebreak [0] 
[N_{3} 
] , \linebreak [0] 
[\epsilon
] , \linebreak [0] 
[\epsilon 1
] , \linebreak [0] 
[\epsilon 2
] , \linebreak [0] 
[FX
] , \linebreak [0] 
[FY
] , \linebreak [0] 
[FZ
] , \linebreak [0] 
[FU
] , \linebreak [0] 
[FV
] , \linebreak [0] 
[FW
] , \linebreak [0] 
[FEP
] , \linebreak [0] 
[RX
] , \linebreak [0] 
[RY
] , \linebreak [0] 
[RZ
] , \linebreak [0] 
[RU
] , \linebreak [0] 
[(SX)
] , \linebreak [0] 
[(SX1)
] , \linebreak [0] 
[(SY)
] , \linebreak [0] 
[(SY1)
] , \linebreak [0] 
[(SZ)
] , \linebreak [0] 
[(SZ1)
] , \linebreak [0] 
[(SU)
] , \linebreak [0] 
[(SU1)
] , \linebreak [0] 
[FXS
] , \linebreak [0] 
[FYS
] , \linebreak [0] 
[(F1)
] , \linebreak [0] 
[(F2)
] , \linebreak [0] 
[(F3)
] , \linebreak [0] 
[(F4)
] , \linebreak [0] 
[(OP1)
] , \linebreak [0] 
[(OP2)
] , \linebreak [0] 
[(R1)
] , \linebreak [0] 
[(S1)
] , \linebreak [0] 
[(S2)
] , \linebreak [0] 
[(EPob)
] , \linebreak [0] 
[(CRS1ob)
] , \linebreak [0] 
[(F1ob)
] , \linebreak [0] 
[(F2ob)
] , \linebreak [0] 
[(F3ob)
] , \linebreak [0] 
[(F4ob)
] , \linebreak [0] 
[(N1ob)
] , \linebreak [0] 
[(N2ob)
] , \linebreak [0] 
[(OP1ob)
] , \linebreak [0] 
[(OP2ob)
] , \linebreak [0] 
[(R1ob)
] , \linebreak [0] 
[(S1ob)
] , \linebreak [0] 
[(S2ob)
] , \linebreak [0] 
[ph_{4}
] , \linebreak [0] 
[ph_{5}
] , \linebreak [0] 
[ph_{6}
] , \linebreak [0] 
[NAT
] , \linebreak [0] 
[RATIONAL_SERIES
] , \linebreak [0] 
[SERIES
] , \linebreak [0] 
[SetOfReals
] , \linebreak [0] 
[SetOfFxs
] , \linebreak [0] 
[N
] , \linebreak [0] 
[Q
] , \linebreak [0] 
[X
] , \linebreak [0] 
[xs
] , \linebreak [0] 
[xaF
] , \linebreak [0] 
[ysF
] , \linebreak [0] 
[us
] , \linebreak [0] 
[usFoelge
] , \linebreak [0] 
[0
] , \linebreak [0] 
[1
] , \linebreak [0] 
[(-1)
] , \linebreak [0] 
[2
] , \linebreak [0] 
[3
] , \linebreak [0] 
[1/2
] , \linebreak [0] 
[1/3
] , \linebreak [0] 
[2/3
] , \linebreak [0] 
[0f
] , \linebreak [0] 
[
\mbox{1f}
] , \linebreak [0] 
[00
] , \linebreak [0] 
[
\mbox{01}
] , \linebreak [0] 
[(--01)
] , \linebreak [0] 
[02
] , \linebreak [0] 
[01//02
] , \linebreak [0] 
[PlusAssociativity(R)
] , \linebreak [0] 
[PlusAssociativity(R)XX
] , \linebreak [0] 
[Plus0(R)
] , \linebreak [0] 
[Negative(R)
] , \linebreak [0] 
[Times1(R)
] , \linebreak [0] 
[lessAddition(R)
] , \linebreak [0] 
[PlusCommutativity(R)
] , \linebreak [0] 
[LeqAntisymmetry(R)
] , \linebreak [0] 
[LeqTransitivity(R)
] , \linebreak [0] 
[leqAddition(R)
] , \linebreak [0] 
[Distribution(R)
] , \linebreak [0] 
[A4(Axiom)
] , \linebreak [0] 
[InductionAxiom
] , \linebreak [0] 
[EqualityAxiom
] , \linebreak [0] 
[EqLeqAxiom
] , \linebreak [0] 
[EqAdditionAxiom
] , \linebreak [0] 
[EqMultiplicationAxiom
] , \linebreak [0] 
[QisClosed(Reciprocal)(Imply)
] , \linebreak [0] 
[QisClosed(Reciprocal)
] , \linebreak [0] 
[QisClosed(Negative)(Imply)
] , \linebreak [0] 
[QisClosed(Negative)
] , \linebreak [0] 
[leqReflexivity
] , \linebreak [0] 
[leqAntisymmetryAxiom
] , \linebreak [0] 
[leqTransitivityAxiom
] , \linebreak [0] 
[leqTotality
] , \linebreak [0] 
[leqAdditionAxiom
] , \linebreak [0] 
[leqMultiplicationAxiom
] , \linebreak [0] 
[plusAssociativity
] , \linebreak [0] 
[plusCommutativity
] , \linebreak [0] 
[Negative
] , \linebreak [0] 
[plus0
] , \linebreak [0] 
[timesAssociativity
] , \linebreak [0] 
[timesCommutativity
] , \linebreak [0] 
[ReciprocalAxiom
] , \linebreak [0] 
[times1
] , \linebreak [0] 
[Distribution
] , \linebreak [0] 
[0not1
] , \linebreak [0] 
[
\mbox{lemma eqLeq(R)}
] , \linebreak [0] 
[TimesAssociativity(R)
] , \linebreak [0] 
[TimesCommutativity(R)
] , \linebreak [0] 
[(Adgic)SameR
] , \linebreak [0] 
[Separation2formula(1)
] , \linebreak [0] 
[Separation2formula(2)
] , \linebreak [0] 
[Cauchy
] , \linebreak [0] 
[PlusF
] , \linebreak [0] 
[ReciprocalF
] , \linebreak [0] 
[From==
] , \linebreak [0] 
[To==
] , \linebreak [0] 
[FromInR
] , \linebreak [0] 
[PlusR(Sym)
] , \linebreak [0] 
[ReciprocalR(Axiom)
] , \linebreak [0] 
[LessMinus1(N)
] , \linebreak [0] 
[Nonnegative(N)
] , \linebreak [0] 
[US0
] , \linebreak [0] 
[NextXS(UpperBound)
] , \linebreak [0] 
[NextXS(NoUpperBound)
] , \linebreak [0] 
[NextUS(UpperBound)
] , \linebreak [0] 
[NextUS(NoUpperBound)
] , \linebreak [0] 
[ExpZero
] , \linebreak [0] 
[ExpPositive
] , \linebreak [0] 
[ExpZero(R)
] , \linebreak [0] 
[ExpPositive(R)
] , \linebreak [0] 
[BSzero
] , \linebreak [0] 
[BSpositive
] , \linebreak [0] 
[UStelescope(Zero)
] , \linebreak [0] 
[UStelescope(Positive)
] , \linebreak [0] 
[EqAddition(R)
] , \linebreak [0] 
[FromLimit
] , \linebreak [0] 
[ToUpperBound
] , \linebreak [0] 
[FromUpperBound
] , \linebreak [0] 
[USisUpperBound
] , \linebreak [0] 
[0not1(R)
] , \linebreak [0] 
[ExpUnbounded(R)
] , \linebreak [0] 
[FromLeq(Advanced)(N)
] , \linebreak [0] 
[FromLeastUpperBound
] , \linebreak [0] 
[ToLeastUpperBound
] , \linebreak [0] 
[XSisNotUpperBound
] , \linebreak [0] 
[ysFGreater
] , \linebreak [0] 
[ysFLess
] , \linebreak [0] 
[SmallInverse
] , \linebreak [0] 
[NatType
] , \linebreak [0] 
[RationalType
] , \linebreak [0] 
[SeriesType
] , \linebreak [0] 
[Max
] , \linebreak [0] 
[Numerical
] , \linebreak [0] 
[NumericalF
] , \linebreak [0] 
[MemberOfSeries(Imply)
] , \linebreak [0] 
[JoinConjuncts(2conditions)
] , \linebreak [0] 
[
\mbox{prop lemma imply negation}
] , \linebreak [0] 
[TND
] , \linebreak [0] 
[FromNegatedImply
] , \linebreak [0] 
[ToNegatedImply
] , \linebreak [0] 
[FromNegated(2*Imply)
] , \linebreak [0] 
[FromNegatedAnd
] , \linebreak [0] 
[FromNegatedOr
] , \linebreak [0] 
[ToNegatedOr
] , \linebreak [0] 
[FromNegations
] , \linebreak [0] 
[From3Disjuncts
] , \linebreak [0] 
[From2*2Disjuncts
] , \linebreak [0] 
[NegateDisjunct1
] , \linebreak [0] 
[NegateDisjunct2
] , \linebreak [0] 
[ExpandDisjuncts
] , \linebreak [0] 
[SENC1
] , \linebreak [0] 
[SENC2
] , \linebreak [0] 
[LessLeq(R)
] , \linebreak [0] 
[MemberOfSeries
] , \linebreak [0] 
[memberOfSeries(Type)
] , \linebreak [0] 
[sup
] , \linebreak [0] 
[ToNegatedAnd(1)
] , \linebreak [0] 
[UniqueNegative
] , \linebreak [0] 
[DoubleMinus
] , \linebreak [0] 
[MinusNegated
] , \linebreak [0] 
[eqReflexivity
] , \linebreak [0] 
[eqSymmetry
] , \linebreak [0] 
[eqTransitivity
] , \linebreak [0] 
[eqTransitivity4
] , \linebreak [0] 
[eqTransitivity5
] , \linebreak [0] 
[eqTransitivity6
] , \linebreak [0] 
[AddEquations
] , \linebreak [0] 
[SubtractEquations
] , \linebreak [0] 
[SubtractEquationsLeft
] , \linebreak [0] 
[MultiplyEquations
] , \linebreak [0] 
[EqNegated
] , \linebreak [0] 
[PositiveToRight(Eq)
] , \linebreak [0] 
[PositiveToLeft(Eq)(1 term)
] , \linebreak [0] 
[NegativeToLeft(Eq)
] , \linebreak [0] 
[NonreciprocalToRight(Eq)(1 term)
] , \linebreak [0] 
[PlusAssociativity(4 terms)
] , \linebreak [0] 
[LessNeq
] , \linebreak [0] 
[NeqSymmetry
] , \linebreak [0] 
[NeqNegated
] , \linebreak [0] 
[SubNeqRight
] , \linebreak [0] 
[SubNeqLeft
] , \linebreak [0] 
[NegativeToRight(Neq)(1 term)
] , \linebreak [0] 
[NeqAddition
] , \linebreak [0] 
[NeqMultiplication
] , \linebreak [0] 
[NonzeroProduct(2)
] , \linebreak [0] 
[UStelescope(+1)
] , \linebreak [0] 
[TelescopeBound(Base)
] , \linebreak [0] 
[TelescopeBound(Indu)
] , \linebreak [0] 
[TelescopeBound
] , \linebreak [0] 
[IntervalSize(Base)
] , \linebreak [0] 
[IntervalSize(Indu)
] , \linebreak [0] 
[IntervalSize
] , \linebreak [0] 
[XS<US
] , \linebreak [0] 
[
\mbox{lemma USdecreasing(+1)}
] , \linebreak [0] 
[CloseUS
] , \linebreak [0] 
[CloseUS(n+1)
] , \linebreak [0] 
[AllNegated(Imply)
] , \linebreak [0] 
[ExistNegated(Imply)
] , \linebreak [0] 
[IntroExist(Helper)
] , \linebreak [0] 
[IntroExist
] , \linebreak [0] 
[ExistMP
] , \linebreak [0] 
[ExistMP2
] , \linebreak [0] 
[TwiceExistMP
] , \linebreak [0] 
[TwiceExistMP2
] , \linebreak [0] 
[EAE-MP
] , \linebreak [0] 
[AddAll 
] , \linebreak [0] 
[AddExist(Helper1)
] , \linebreak [0] 
[AddExist(Helper2)
] , \linebreak [0] 
[AddExist
] , \linebreak [0] 
[AddExist(SimpleAnt)
] , \linebreak [0] 
[AddExist(Simple)
] , \linebreak [0] 
[AddEAE
] , \linebreak [0] 
[AEA-negated
] , \linebreak [0] 
[EEA-negated
] , \linebreak [0] 
[Induction
] , \linebreak [0] 
[leqAntisymmetry
] , \linebreak [0] 
[leqTransitivity
] , \linebreak [0] 
[leqAddition
] , \linebreak [0] 
[leqMultiplication
] , \linebreak [0] 
[Reciprocal
] , \linebreak [0] 
[Equality
] , \linebreak [0] 
[eqLeq
] , \linebreak [0] 
[eqAddition
] , \linebreak [0] 
[eqMultiplication
] , \linebreak [0] 
[LeqMultiplicationLeft 
] , \linebreak [0] 
[LeqLessEq
] , \linebreak [0] 
[LessLeq
] , \linebreak [0] 
[FromLeqGeq
] , \linebreak [0] 
[subLeqRight
] , \linebreak [0] 
[subLeqLeft
] , \linebreak [0] 
[Leq+1
] , \linebreak [0] 
[PositiveToRight(Leq)
] , \linebreak [0] 
[PositiveToRight(Leq)(1 term)
] , \linebreak [0] 
[
\mbox{lemma negativeToRight(Leq)}
] , \linebreak [0] 
[PositiveToLeft(Leq)
] , \linebreak [0] 
[negativeToLeft(Leq)
] , \linebreak [0] 
[negativeToLeft(Leq)(1 term)
] , \linebreak [0] 
[LeqAdditionLeft
] , \linebreak [0] 
[leqSubtraction
] , \linebreak [0] 
[leqSubtractionLeft
] , \linebreak [0] 
[thirdGeq
] , \linebreak [0] 
[LeqNegated
] , \linebreak [0] 
[AddEquations(Leq)
] , \linebreak [0] 
[MultiplyEquations(Leq)
] , \linebreak [0] 
[ThirdGeqSeries
] , \linebreak [0] 
[LeqNeqLess
] , \linebreak [0] 
[FromLess
] , \linebreak [0] 
[ToLess
] , \linebreak [0] 
[fromNotLess
] , \linebreak [0] 
[toNotLess
] , \linebreak [0] 
[NegativeLessPositive
] , \linebreak [0] 
[leqLessTransitivity
] , \linebreak [0] 
[LessLeqTransitivity
] , \linebreak [0] 
[LessTransitivity
] , \linebreak [0] 
[LessTotality
] , \linebreak [0] 
[SubLessRight
] , \linebreak [0] 
[SubLessLeft
] , \linebreak [0] 
[SwitchTerms(x<y-z)
] , \linebreak [0] 
[SwitchTerms(x-y<z)
] , \linebreak [0] 
[LessAddition
] , \linebreak [0] 
[LessAdditionLeft
] , \linebreak [0] 
[LessMultiplication
] , \linebreak [0] 
[LessMultiplicationLeft
] , \linebreak [0] 
[LessDivision
] , \linebreak [0] 
[PositiveToRight(Less)
] , \linebreak [0] 
[PositiveToLeft(Less)
] , \linebreak [0] 
[NegativeToLeft(Less)
] , \linebreak [0] 
[NegativeToRight(Less)
] , \linebreak [0] 
[AddEquations(Less)
] , \linebreak [0] 
[AddEquations(LeqLess)
] , \linebreak [0] 
[reciprocalToLeft(Less)
] , \linebreak [0] 
[LessNegated
] , \linebreak [0] 
[PositiveNonzero
] , \linebreak [0] 
[PositiveNegated
] , \linebreak [0] 
[NonpositiveNegated
] , \linebreak [0] 
[NegativeNegated
] , \linebreak [0] 
[NonnegativeNegated
] , \linebreak [0] 
[PositiveHalved
] , \linebreak [0] 
[PositiveInverted
] , \linebreak [0] 
[NonnegativeNumerical
] , \linebreak [0] 
[NegativeNumerical
] , \linebreak [0] 
[PositiveNumerical
] , \linebreak [0] 
[
\mbox{lemma nonpositiveNumerical}
] , \linebreak [0] 
[|0|=0
] , \linebreak [0] 
[0<=|x|
] , \linebreak [0] 
[x<=|x|
] , \linebreak [0] 
[FromPositiveNumerical
] , \linebreak [0] 
[SameNumerical
] , \linebreak [0] 
[SignNumerical(+)
] , \linebreak [0] 
[SignNumerical
] , \linebreak [0] 
[ToNumericalLess
] , \linebreak [0] 
[FromNumericalGreater
] , \linebreak [0] 
[NumericalDifference
] , \linebreak [0] 
[NumericalDifferenceLess(Helper)
] , \linebreak [0] 
[NumericalDifferenceLess
] , \linebreak [0] 
[SplitNumericalSumHelper
] , \linebreak [0] 
[splitNumericalSum(++)
] , \linebreak [0] 
[splitNumericalSum(--)
] , \linebreak [0] 
[splitNumericalSum(+-small)
] , \linebreak [0] 
[splitNumericalSum(+-big)
] , \linebreak [0] 
[splitNumericalSum(+-)
] , \linebreak [0] 
[splitNumericalSum(-+)
] , \linebreak [0] 
[splitNumericalSum
] , \linebreak [0] 
[SplitNumericalProduct(++)
] , \linebreak [0] 
[SplitNumericalProduct(+-)
] , \linebreak [0] 
[SplitNumericalProduct
] , \linebreak [0] 
[insertMiddleTerm(Numerical)
] , \linebreak [0] 
[insertTwoMiddleTerms(Numerical)
] , \linebreak [0] 
[Three2twoTerms
] , \linebreak [0] 
[Three2threeTerms
] , \linebreak [0] 
[Three2twoFactors
] , \linebreak [0] 
[Three2threeFactors
] , \linebreak [0] 
[Times(-1)
] , \linebreak [0] 
[Times(-1)Left
] , \linebreak [0] 
[MaxLeq(1)
] , \linebreak [0] 
[MaxLeq(2)
] , \linebreak [0] 
[LessThanMax
] , \linebreak [0] 
[x+y=zBackwards
] , \linebreak [0] 
[x*y=zBackwards
] , \linebreak [0] 
[x=x+(y-y)
] , \linebreak [0] 
[x=x+y-y
] , \linebreak [0] 
[x=x*y*(1/y)
] , \linebreak [0] 
[insertMiddleTerm(Sum)
] , \linebreak [0] 
[insertTwoMiddleTerms(Sum)
] , \linebreak [0] 
[insertMiddleTerm(Difference)
] , \linebreak [0] 
[x*0+x=x
] , \linebreak [0] 
[x*0=0
] , \linebreak [0] 
[NonnegativeFactors
] , \linebreak [0] 
[NonzeroFactors
] , \linebreak [0] 
[PositiveFactors
] , \linebreak [0] 
[PlusTimesMinus
] , \linebreak [0] 
[MinusTimesMinus
] , \linebreak [0] 
[(-1)*(-1)+(-1)*1=0
] , \linebreak [0] 
[(-1)*(-1)=1
] , \linebreak [0] 
[0<1Helper
] , \linebreak [0] 
[0<1
] , \linebreak [0] 
[0<2
] , \linebreak [0] 
[0<3
] , \linebreak [0] 
[0<1/2
] , \linebreak [0] 
[0<1/3
] , \linebreak [0] 
[TwoWholes
] , \linebreak [0] 
[ThreeWholes
] , \linebreak [0] 
[TwoHalves
] , \linebreak [0] 
[ThreeThirds
] , \linebreak [0] 
[-x-y=-(x+y)
] , \linebreak [0] 
[-x*y=-(x*y)
] , \linebreak [0] 
[-0=0
] , \linebreak [0] 
[SFsymmetry
] , \linebreak [0] 
[SFtransitivity
] , \linebreak [0] 
[f2R(Plus)
] , \linebreak [0] 
[f2R(Times)
] , \linebreak [0] 
[<<TransitivityHelper(Q)
] , \linebreak [0] 
[<<Transitivity
] , \linebreak [0] 
[<<==Reflexivity
] , \linebreak [0] 
[<<==AntisymmetryHelper(Q)
] , \linebreak [0] 
[FromNot<f(Weak)(Helper)
] , \linebreak [0] 
[FromNot<f(Weak)
] , \linebreak [0] 
[FromNot<f(Strong)(Helper2)
] , \linebreak [0] 
[FromNot<f(Strong)(Helper)
] , \linebreak [0] 
[FromNot<f(Strong)
] , \linebreak [0] 
[fromNotSameF(Strongest)(Helper2)
] , \linebreak [0] 
[fromNotSameF(Strongest)(Helper)
] , \linebreak [0] 
[fromNotSameF(Strongest)
] , \linebreak [0] 
[ToLess(F)(Helper)
] , \linebreak [0] 
[ToLess(F)
] , \linebreak [0] 
[FromNot<<
] , \linebreak [0] 
[ToLess(R)
] , \linebreak [0] 
[FromNotSameF(Weak)(Helper)
] , \linebreak [0] 
[FromNotSameF(Weak)
] , \linebreak [0] 
[FromNotLess(F)
] , \linebreak [0] 
[==Addition
] , \linebreak [0] 
[==AdditionLeft
] , \linebreak [0] 
[Fpart-Bounded(Base)
] , \linebreak [0] 
[Fpart-Bounded(InduHelper)
] , \linebreak [0] 
[Fpart-Bounded(Indu)
] , \linebreak [0] 
[Fpart-Bounded
] , \linebreak [0] 
[F-Bounded(Helper)
] , \linebreak [0] 
[F-Bounded
] , \linebreak [0] 
[SameFmultiplication(Helper)
] , \linebreak [0] 
[SameFmultiplication
] , \linebreak [0] 
[EqMultiplication(R)
] , \linebreak [0] 
[EqMultiplicationLeft(R)
] , \linebreak [0] 
[x*0=0(F)
] , \linebreak [0] 
[x*0=0(R)
] , \linebreak [0] 
[LessMultiplication(F)(Helper2)
] , \linebreak [0] 
[LessMultiplication(F)(Helper)
] , \linebreak [0] 
[LessMultiplication(F)
] , \linebreak [0] 
[LessMultiplication(R)
] , \linebreak [0] 
[LeqMultiplication(R)
] , \linebreak [0] 
[PlusAssociativity(F)
] , \linebreak [0] 
[Plus0(F)
] , \linebreak [0] 
[PlusCommutativity(F)
] , \linebreak [0] 
[TimesAssociativity(F)
] , \linebreak [0] 
[Times1f
] , \linebreak [0] 
[Cauchy(2)(Helper)
] , \linebreak [0] 
[Cauchy(2)
] , \linebreak [0] 
[ReciprocalFnonzero
] , \linebreak [0] 
[(Eventually=f)2sameF(Helper)
] , \linebreak [0] 
[(Eventually=f)2sameF
] , \linebreak [0] 
[FromNotSameF(Strong)(Helper2)
] , \linebreak [0] 
[FromNotSameF(Strong)(Helper)
] , \linebreak [0] 
[FromNotSameF(Strong)
] , \linebreak [0] 
[SameFreciprocal(Helper)
] , \linebreak [0] 
[SameFreciprocal
] , \linebreak [0] 
[From!!==
] , \linebreak [0] 
[Reciprocal(R)
] , \linebreak [0] 
[TimesCommutativity(F)
] , \linebreak [0] 
[Distribution(F)
] , \linebreak [0] 
[FromMax(1)
] , \linebreak [0] 
[FromMax(2)
] , \linebreak [0] 
[ToNegatedAnd
] , \linebreak [0] 
[DistributionOut
] , \linebreak [0] 
[DistributionOutLeft
] , \linebreak [0] 
[DistributionLeft
] , \linebreak [0] 
[FromNotLess(R)
] , \linebreak [0] 
[CartProdIsRelation
] , \linebreak [0] 
[FromSubset
] , \linebreak [0] 
[SubsetIsRelation
] , \linebreak [0] 
[ToSeries
] , \linebreak [0] 
[FromSeries
] , \linebreak [0] 
[SeriesSubsetCP
] , \linebreak [0] 
[ValueType
] , \linebreak [0] 
[RemoveOr
] , \linebreak [0] 
[FromSingleton
] , \linebreak [0] 
[InPair(1)
] , \linebreak [0] 
[InPair(2)
] , \linebreak [0] 
[SameMember(2)
] , \linebreak [0] 
[ToBinaryUnion(1)
] , \linebreak [0] 
[ToBinaryUnion(2)
] , \linebreak [0] 
[FromOrderedPair(TwoLevels)
] , \linebreak [0] 
[ToCartProd(Helper)
] , \linebreak [0] 
[ToCartProd
] , \linebreak [0] 
[NonreciprocalToRight(Eq)
] , \linebreak [0] 
[NonreciprocalToLeft(Eq)(1 term)
] , \linebreak [0] 
[SameReciprocal
] , \linebreak [0] 
[CPseparationIsRelation
] , \linebreak [0] 
[OrderedPairEquality
] , \linebreak [0] 
[ReciprocalIsFunction
] , \linebreak [0] 
[ReciprocalIsTotal
] , \linebreak [0] 
[ReciprocalIsRationalSeries
] , \linebreak [0] 
[CrsIsRelation
] , \linebreak [0] 
[CrsIsFunction 
] , \linebreak [0] 
[CrsIsTotal
] , \linebreak [0] 
[CrsIsSeries
] , \linebreak [0] 
[CrsLookup
] , \linebreak [0] 
[0f
] , \linebreak [0] 
[1f
] , \linebreak [0] 
[ToSingleton
] , \linebreak [0] 
[FromSameSingleton
] , \linebreak [0] 
[SingletonmembersEqual
] , \linebreak [0] 
[UnequalsNotInSingleton
] , \linebreak [0] 
[NonsingletonmembersUnequal
] , \linebreak [0] 
[FromOrderedPair
] , \linebreak [0] 
[FromOrderedPair(1)
] , \linebreak [0] 
[FromOrderedPair(2)
] , \linebreak [0] 
[FromCartProd
] , \linebreak [0] 
[FromCartProd(1)
] , \linebreak [0] 
[FromCartProd(2)
] , \linebreak [0] 
[sameOrderedPair
] , \linebreak [0] 
[InSeriesHelper
] , \linebreak [0] 
[InSeries
] , \linebreak [0] 
[To=f(Subset)(Helper)
] , \linebreak [0] 
[To=f(Subset)
] , \linebreak [0] 
[To=f
] , \linebreak [0] 
[productIsFunction
] , \linebreak [0] 
[productIsTotal
] , \linebreak [0] 
[ProductIsRationalSeries
] , \linebreak [0] 
[TimesF
] , \linebreak [0] 
[-x+(1/2)x=-(1/2)x
] , \linebreak [0] 
[PositiveTripled
] , \linebreak [0] 
[PositiveDividedBy3
] , \linebreak [0] 
[|x-x|=0
] , \linebreak [0] 
[1<2
] , \linebreak [0] 
[1/3<2/3
] , \linebreak [0] 
[(1/3)x+(1/3)x=(2/3)x
] , \linebreak [0] 
[(2/3)x+(1/3)x=x
] , \linebreak [0] 
[-x+(2/3)x=-(1/3)x
] , \linebreak [0] 
[-(1/3)x-(1/3)x=-(2/3)x
] , \linebreak [0] 
[-x+(1/3)x=-(2/3)x
] , \linebreak [0] 
[PreserveLessGreater
] , \linebreak [0] 
[ClosetolessIsLess
] , \linebreak [0] 
[SubLessLeft(F)
] , \linebreak [0] 
[SubLessLeft(R)
] , \linebreak [0] 
[ClosetogreaterIsGreater
] , \linebreak [0] 
[SubLessRight(F)
] , \linebreak [0] 
[SubLessRight(R)
] , \linebreak [0] 
[plus0Left
] , \linebreak [0] 
[times1Left
] , \linebreak [0] 
[EqAdditionLeft
] , \linebreak [0] 
[EqMultiplicationLeft
] , \linebreak [0] 
[PlusF(Sym)
] , \linebreak [0] 
[TimesF(Sym)
] , \linebreak [0] 
[SameSeries(Gen)
] , \linebreak [0] 
[EqualsSameF
] , \linebreak [0] 
[LeqReflexivity(R)
] , \linebreak [0] 
[LeqTotality(R)
] , \linebreak [0] 
[PositiveToLeft(Eq)
] , \linebreak [0] 
[ExpZero(Exact) 
] , \linebreak [0] 
[SameExp(Base)
] , \linebreak [0] 
[SameExp(Indu)
] , \linebreak [0] 
[SameExp
] , \linebreak [0] 
[(1/2)(x+y)-x=(1/2)(y-x)
] , \linebreak [0] 
[y-(1/2)(x+y)=(1/2)(y-x)
] , \linebreak [0] 
[BSzero(Exact)
] , \linebreak [0] 
[SameBS(2)(Base)
] , \linebreak [0] 
[SameBS(2)(Indu)
] , \linebreak [0] 
[SameBS(2)
] , \linebreak [0] 
[NegativeToLeft(Less)(1 term)
] , \linebreak [0] 
[BS(+1)
] , \linebreak [0] 
[BSbound(Exact)(Base)
] , \linebreak [0] 
[BSbound(Exact)(Indu)
] , \linebreak [0] 
[BSbound(Exact)
] , \linebreak [0] 
[BSbound
] , \linebreak [0] 
[UStelescope(Zero)(Exact)
] , \linebreak [0] 
[SameTelescope(2)(Base)
] , \linebreak [0] 
[SameTelescope(2)(Indu)
] , \linebreak [0] 
[SameTelescope(2)
] , \linebreak [0] 
[Exp(+1)
] , \linebreak [0] 
[PositiveBase(Base)
] , \linebreak [0] 
[PositiveBase(Indu)
] , \linebreak [0] 
[PositiveBase
] , \linebreak [0] 
[TelescopeNumerical(Base)
] , \linebreak [0] 
[TelescopeNumerical(Indu)
] , \linebreak [0] 
[TelescopeNumerical
] , \linebreak [0] 
[(+1)IsPositive(N)
] , \linebreak [0] 
[DistributionOut(Minus)
] , \linebreak [0] 
[PositiveToRight(Eq)(1 term)
] , \linebreak [0] 
[SameSeries(NumDiff)
] , \linebreak [0] 
[ToNegatedDoubleImply
] , \linebreak [0] 
[AddNegatedAll
] , \linebreak [0] 
[(A)to(~E~)(Imply)
] , \linebreak [0] 
[(E)to(~A~)(Imply)
] , \linebreak [0] 
[(E~)to(~A)(Imply)
] , \linebreak [0] 
[ToNegatedAEA 
] , \linebreak [0] 
[Three2threeTerms(R)
] , \linebreak [0] 
[LessNeq(F)(Helper)
] , \linebreak [0] 
[LessNeq(F)
] , \linebreak [0] 
[LessNeq(R)
] , \linebreak [0] 
[x=x+(y-y)(R)
] , \linebreak [0] 
[x=x+y-y(R)
] , \linebreak [0] 
[SubtractEquations(R)
] , \linebreak [0] 
[NeqAddition(R)
] , \linebreak [0] 
[PositiveToRight(Less)(R)
] , \linebreak [0] 
[PositiveToRight(Less)(1 term)(R)
] , \linebreak [0] 
[LeqNeqLess(R)
] , \linebreak [0] 
[SubLeqLeft(R)
] , \linebreak [0] 
[ToLeq(Advanced)(R)
] , \linebreak [0] 
[LeqLessTransitivity(R)
] , \linebreak [0] 
[NegativeToLeft(Eq)(R)
] , \linebreak [0] 
[NegativeToRight(Less)(R)
] , \linebreak [0] 
[!!==Symmetry
] , \linebreak [0] 
[SwitchTerms(x<=y-z)
] , \linebreak [0] 
[Plus0Left(R)
] , \linebreak [0] 
[PositiveToRight(Eq)(R)
] , \linebreak [0] 
[EqAdditionLeft(R)
] , \linebreak [0] 
[Three2twoTerms(R)
] , \linebreak [0] 
[To!!==
] , \linebreak [0] 
[PositiveToRight(Less)(1 term)
] , \linebreak [0] 
[(A~)to(~E)
] , \linebreak [0] 
[NegativeToRight(Eq)(R)
] , \linebreak [0] 
[NegativeToRight(Eq)(1 term)(R)
] , \linebreak [0] 
[DoubleMinus(R)
] , \linebreak [0] 
[UniqueNegative(R)
] , \linebreak [0] 
[SubtractEquationsLeft(R)
] , \linebreak [0] 
[EqNegated(R)
] , \linebreak [0] 
[NeqNegated(R)
] , \linebreak [0] 
[-0=0(R)
] , \linebreak [0] 
[NegativeNegated(R)
] , \linebreak [0] 
[FromLeqGeq(R)
] , \linebreak [0] 
[0<=|x|(R)
] , \linebreak [0] 
[PositiveNegated(R)
] , \linebreak [0] 
[AddEquations(R)
] , \linebreak [0] 
[Times(-1)(R)
] , \linebreak [0] 
[Times(-1)Left(R)
] , \linebreak [0] 
[-x-y=-(x+y)(R)
] , \linebreak [0] 
[LessTotality(R)
] , \linebreak [0] 
[SameNumerical(R)
] , \linebreak [0] 
[MinusNegated(R)
] , \linebreak [0] 
[PositiveNumerical(R)
] , \linebreak [0] 
[SignNumerical(+)(R)
] , \linebreak [0] 
[NonnegativeNumerical(R)
] , \linebreak [0] 
[NegativeNumerical(R)
] , \linebreak [0] 
[LeqNegated(R)
] , \linebreak [0] 
[LessNegated(R)
] , \linebreak [0] 
[SubLeqRight(R)
] , \linebreak [0] 
[FromLess(R)
] , \linebreak [0] 
[DistributionOut(R)
] , \linebreak [0] 
[x*0+x=x(R)
] , \linebreak [0] 
[x*0=0(R)(fff)
] , \linebreak [0] 
[SignNumerical(R)
] , \linebreak [0] 
[NumericalDifference(R)
] , \linebreak [0] 
[x<=|x|(R)
] , \linebreak [0] 
[USlimitIsUpperBound(Helper)
] , \linebreak [0] 
[USlimitIsUpperBound
] , \linebreak [0] 
[(-1)*(-1)+(-1)*1=0(R)
] , \linebreak [0] 
[(-1)*(-1)=1(R)
] , \linebreak [0] 
[0<1Helper(R)
] , \linebreak [0] 
[0<1(R)
] , \linebreak [0] 
[ExpZero(Exact)(R)
] , \linebreak [0] 
[PositiveBase(R)(Base)
] , \linebreak [0] 
[Three2twoFactors(R)
] , \linebreak [0] 
[x=x*y*(1/y)(R)
] , \linebreak [0] 
[NeqMultiplication(R)
] , \linebreak [0] 
[LessTransitivity(R)
] , \linebreak [0] 
[0<2(R)
] , \linebreak [0] 
[SameExp(R)(Base)
] , \linebreak [0] 
[SameExp(R)(Indu)
] , \linebreak [0] 
[SameExp(R)
] , \linebreak [0] 
[SubNeqLeft(R)
] , \linebreak [0] 
[SubNeqRight(R)
] , \linebreak [0] 
[NonzeroFactors(R)
] , \linebreak [0] 
[NonnegativeFactors(R)
] , \linebreak [0] 
[PositiveFactors(R)
] , \linebreak [0] 
[LessDivision(R)
] , \linebreak [0] 
[0<1/2(R)
] , \linebreak [0] 
[PositiveToRight(Eq)(1 term)(R)
] , \linebreak [0] 
[Exp(+1)(R)
] , \linebreak [0] 
[PositiveBase(R)(Indu)
] , \linebreak [0] 
[PositiveBase(R)
] , \linebreak [0] 
[-x*y=-(x*y)(R)
] , \linebreak [0] 
[PositiveToLeft(Eq)(R)
] , \linebreak [0] 
[Times1Left(R)
] , \linebreak [0] 
[x+x=2*x(R)
] , \linebreak [0] 
[(1/2)x+(1/2)x=x(R)
] , \linebreak [0] 
[DistributionOut(Minus)(R)
] , \linebreak [0] 
[(1/2)(x+y)-x=(1/2)(y-x)(R)
] , \linebreak [0] 
[IntervalSize(R)(Base)
] , \linebreak [0] 
[LessMultiplicationLeft(R)
] , \linebreak [0] 
[NegativeToLeft(Less)(R)
] , \linebreak [0] 
[NegativeToLeft(Less)(1 term)(R)
] , \linebreak [0] 
[y-(1/2)(x+y)=(1/2)(y-x)(R)
] , \linebreak [0] 
[IntervalSize(R)(Indu)
] , \linebreak [0] 
[IntervalSize(R)
] , \linebreak [0] 
[XSlessUS(R)
] , \linebreak [0] 
[USdecreasing(+1)(R)
] , \linebreak [0] 
[ExpUnbounded(Base)
] , \linebreak [0] 
[ExpUnbounded(Indu)
] , \linebreak [0] 
[ExpUnbounded
] , \linebreak [0] 
[1<=x+1(N)
] , \linebreak [0] 
[ExpNonzero(Base)
] , \linebreak [0] 
[ExpNonzero(Indu)
] , \linebreak [0] 
[ExpNonzero
] , \linebreak [0] 
[ExpNonzero(2)
] , \linebreak [0] 
[HalfBase(Base)
] , \linebreak [0] 
[HalfBase(Indu)
] , \linebreak [0] 
[MultiplyEquations(R)
] , \linebreak [0] 
[NonreciprocalToRight(Eq)(1 term)(R)
] , \linebreak [0] 
[PositiveNonzero(R)
] , \linebreak [0] 
[NonzeroProduct(2)(R)
] , \linebreak [0] 
[HalfBase
] , \linebreak [0] 
[Three2threeFactors(R)
] , \linebreak [0] 
[x*y=zBackwards(R)
] , \linebreak [0] 
[PositiveInverted(R)
] , \linebreak [0] 
[ReciprocalToRight(Less)(R)
] , \linebreak [0] 
[ReciprocalToRight(Less)(1 term)(R)
] , \linebreak [0] 
[NonreciprocalToLeft(Less)(R)
] , \linebreak [0] 
[1<x*y(R)
] , \linebreak [0] 
[SwitchFactors(1/x<y)(R)
] , \linebreak [0] 
[SmallHalving
] , \linebreak [0] 
[IntervalSize(anyPositive)
] , \linebreak [0] 
[USdecreasing(+n)(Base)
] , \linebreak [0] 
[USdecreasing(+n)(Indu)
] , \linebreak [0] 
[USdecreasing(+n)
] , \linebreak [0] 
[USdecreasing
] , \linebreak [0] 
[LeqAdditionLeft(R)
] , \linebreak [0] 
[ToNotLess(R)
] , \linebreak [0] 
[LimitOfUSIsLeq
] , \linebreak [0] 
[SubtractEquations(Less)(R)
] , \linebreak [0] 
[SubtractEquationsLeft(Less)(R)
] , \linebreak [0] 
[LessNegated(Negative)(R)
] , \linebreak [0] 
[FromNegatedAnd(Imply)
] , \linebreak [0] 
[RemoveDoubleNeg(Consequent)
] , \linebreak [0] 
[FromNotUpperBound
] , \linebreak [0] 
[LeqNUB
] , \linebreak [0] 
[USlimitIsLeastUpperBound(Helper)
] , \linebreak [0] 
[USlimitIsLeastUpperBound
] , \linebreak [0] 
[ExistMP3
] , \linebreak [0] 
[GreaterPositive(N)
] , \linebreak [0] 
[ysFClose(Helper)
] , \linebreak [0] 
[ysFClose
] , \linebreak [0] 
[ysFCauchy(Helper)
] , \linebreak [0] 
[ysFCauchy
] , \linebreak [0] 
[from<<==
] , \linebreak [0] 
[to<<==
] , \linebreak [0] 
[NonnegativeNumerical(F)
] , \linebreak [0] 
[NegativeNumerical(F)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{tester1}
] , \linebreak [0] 
[
\mbox{tester2}
] , \linebreak [0] 
[
\mbox{tester3}
] , \linebreak [0] 
[
\mbox{tester4}
] , \linebreak [0] 
[
\mbox{tester5}
] , \linebreak [0] 
[
\mbox{tester6}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\_\{
\ast 
\}
] , \linebreak [0] 
[
\ast 
/indexintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/intro(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/bothintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
/nameintro(
\ast 
,
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast '
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
[ 
\ast 
{\Rightarrow} 
\ast 
]
] , \linebreak [0] 
[
\ast 
0
] , \linebreak [0] 
[
\ast 
1
] , \linebreak [0] 
[
0 \mathrm {b}
] , \linebreak [0] 
[
\ast 
\mbox {-color}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
\mbox {-color}^{\ast}( 
\ast 
)
] , \linebreak [0] 
[
\ast 
{}^H
] , \linebreak [0] 
[
\ast 
{}^T
] , \linebreak [0] 
[
\ast 
{}^U
] , \linebreak [0] 
[
\ast 
{}^h
] , \linebreak [0] 
[
\ast 
{}^t
] , \linebreak [0] 
[
\ast 
{}^s
] , \linebreak [0] 
[
\ast 
{}^c
] , \linebreak [0] 
[
\ast 
{}^d
] , \linebreak [0] 
[
\ast 
{}^a
] , \linebreak [0] 
[
\ast 
{}^C
] , \linebreak [0] 
[
\ast 
{}^M
] , \linebreak [0] 
[
\ast 
{}^B
] , \linebreak [0] 
[
\ast 
{}^{r}
] , \linebreak [0] 
[
\ast 
{}^{i}
] , \linebreak [0] 
[
\ast 
{}^{d}
] , \linebreak [0] 
[
\ast 
{}^{R}
] , \linebreak [0] 
[
\ast 
{}^{0}
] , \linebreak [0] 
[
\ast 
{}^{1}
] , \linebreak [0] 
[
\ast 
{}^{2}
] , \linebreak [0] 
[
\ast 
{}^{3}
] , \linebreak [0] 
[
\ast 
{}^{4}
] , \linebreak [0] 
[
\ast 
{}^{5}
] , \linebreak [0] 
[
\ast 
{}^{6}
] , \linebreak [0] 
[
\ast 
{}^{7}
] , \linebreak [0] 
[
\ast 
{}^{8}
] , \linebreak [0] 
[
\ast 
{}^{9}
] , \linebreak [0] 
[
\ast 
{} ^ { E }
] , \linebreak [0] 
[
\ast 
{} ^ {\cal V}
] , \linebreak [0] 
[
\ast 
{} ^ {\cal C}
] , \linebreak [0] 
[
\ast 
{} ^ {{\cal C} ^ { \ast }}
] , \linebreak [0] 
[
\ast 
{}^{hide}
]
; 
\newline \mathbf {Preassociative} \newline 
[
\mbox{``}
\ast 
\mbox{''}
] , \linebreak [0] 
[
] , \linebreak [0] 
[
(
\ast 
)^{\bf t}
] , \linebreak [0] 
[
\mbox{string}(
\ast 
)+
\ast 
] , \linebreak [0] 
[\mbox{string}(
\ast 
)\mathrel{++}
\ast 
] , \linebreak [0] 
[
\newline 
\ast 
] , \linebreak [0] 
[
\linebreak [0]\ \hskip0em plus2.0em{}
\ast 
] , \linebreak [0] 
[!
\ast 
] , \linebreak [0] 
[
\mbox {\tt \char34}
\ast 
] , \linebreak [0] 
[
\#
\ast 
] , \linebreak [0] 
[
\$
\ast 
] , \linebreak [0] 
[
\%
\ast 
] , \linebreak [0] 
[
\&
\ast 
] , \linebreak [0] 
[
\mbox {'}
\ast 
] , \linebreak [0] 
[(
\ast 
] , \linebreak [0] 
[)
\ast 
] , \linebreak [0] 
[
{*}
\ast 
] , \linebreak [0] 
[
{+}
\ast 
] , \linebreak [0] 
[,
\ast 
] , \linebreak [0] 
[
\mbox{-}
\ast 
] , \linebreak [0] 
[.
\ast 
] , \linebreak [0] 
[/
\ast 
] , \linebreak [0] 
[0
\ast 
] , \linebreak [0] 
[1
\ast 
] , \linebreak [0] 
[2
\ast 
] , \linebreak [0] 
[3
\ast 
] , \linebreak [0] 
[4
\ast 
] , \linebreak [0] 
[5
\ast 
] , \linebreak [0] 
[6
\ast 
] , \linebreak [0] 
[7
\ast 
] , \linebreak [0] 
[8
\ast 
] , \linebreak [0] 
[9
\ast 
] , \linebreak [0] 
[
{:}
\ast 
] , \linebreak [0] 
[;
\ast 
] , \linebreak [0] 
[
{<}
\ast 
] , \linebreak [0] 
[
{=}
\ast 
] , \linebreak [0] 
[
{>}
\ast 
] , \linebreak [0] 
[?
\ast 
] , \linebreak [0] 
[@
\ast 
] , \linebreak [0] 
[A
\ast 
] , \linebreak [0] 
[B
\ast 
] , \linebreak [0] 
[C
\ast 
] , \linebreak [0] 
[D
\ast 
] , \linebreak [0] 
[E
\ast 
] , \linebreak [0] 
[F
\ast 
] , \linebreak [0] 
[G
\ast 
] , \linebreak [0] 
[H
\ast 
] , \linebreak [0] 
[I
\ast 
] , \linebreak [0] 
[J
\ast 
] , \linebreak [0] 
[K
\ast 
] , \linebreak [0] 
[L
\ast 
] , \linebreak [0] 
[M
\ast 
] , \linebreak [0] 
[N
\ast 
] , \linebreak [0] 
[O
\ast 
] , \linebreak [0] 
[P
\ast 
] , \linebreak [0] 
[Q
\ast 
] , \linebreak [0] 
[R
\ast 
] , \linebreak [0] 
[S
\ast 
] , \linebreak [0] 
[T
\ast 
] , \linebreak [0] 
[U
\ast 
] , \linebreak [0] 
[V
\ast 
] , \linebreak [0] 
[W
\ast 
] , \linebreak [0] 
[X
\ast 
] , \linebreak [0] 
[Y
\ast 
] , \linebreak [0] 
[Z
\ast 
] , \linebreak [0] 
[[
\ast 
] , \linebreak [0] 
[
\mbox {$\backslash $}
\ast 
] , \linebreak [0] 
[]
\ast 
] , \linebreak [0] 
[
{\char94}
\ast 
] , \linebreak [0] 
[
\_
\ast 
] , \linebreak [0] 
[
\mbox {`}
\ast 
] , \linebreak [0] 
[a
\ast 
] , \linebreak [0] 
[b
\ast 
] , \linebreak [0] 
[c
\ast 
] , \linebreak [0] 
[d
\ast 
] , \linebreak [0] 
[e
\ast 
] , \linebreak [0] 
[f
\ast 
] , \linebreak [0] 
[g
\ast 
] , \linebreak [0] 
[h
\ast 
] , \linebreak [0] 
[i
\ast 
] , \linebreak [0] 
[j
\ast 
] , \linebreak [0] 
[k
\ast 
] , \linebreak [0] 
[l
\ast 
] , \linebreak [0] 
[m
\ast 
] , \linebreak [0] 
[n
\ast 
] , \linebreak [0] 
[o
\ast 
] , \linebreak [0] 
[p
\ast 
] , \linebreak [0] 
[q
\ast 
] , \linebreak [0] 
[r
\ast 
] , \linebreak [0] 
[s
\ast 
] , \linebreak [0] 
[t
\ast 
] , \linebreak [0] 
[u
\ast 
] , \linebreak [0] 
[v
\ast 
] , \linebreak [0] 
[w
\ast 
] , \linebreak [0] 
[x
\ast 
] , \linebreak [0] 
[y
\ast 
] , \linebreak [0] 
[z
\ast 
] , \linebreak [0] 
[
\{
\ast 
] , \linebreak [0] 
[|
\ast 
] , \linebreak [0] 
[
\}
\ast 
] , \linebreak [0] 
[
\char126 
\ast 
] , \linebreak [0] 
[
\mathbf{Preassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
\mathbf{Postassociative}\,
\ast 
; 
\ast 
] , \linebreak [0] 
[
[
\ast 
] , \linebreak [0] 
\ast 
] , \linebreak [0] 
[
\mathrm{priority} \,
\ast 
\, \mathrm{end}
] , \linebreak [0] 
[
newline\ 
\ast 
] , \linebreak [0] 
[
macro\ newline\ 
\ast 
] , \linebreak [0] 
[
MacroIndent(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbin {\mbox {'}}
\ast 
] , \linebreak [0] 
[
\ast 
\mathbin {\mbox {`}}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ 
\ast 
 (exp) 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{}'
] , \linebreak [0] 
[R(
\ast 
)
] , \linebreak [0] 
[--R(
\ast 
)
] , \linebreak [0] 
[rec
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 / 
\ast 
] , \linebreak [0] 
[
\ast 
 \cap 
\ast 
] , \linebreak [0] 
[
\ast 
[
\ast 
]
]
; 
\newline \mathbf {Preassociative} \newline 
[\cup 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\cup} 
\ast 
] , \linebreak [0] 
[P(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
\}
] , \linebreak [0] 
[StateExpand(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[extractSeries(
\ast 
)
] , \linebreak [0] 
[SetOfSeries(
\ast 
)
] , \linebreak [0] 
[--Macro(
\ast 
)
] , \linebreak [0] 
[ExpandList(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[**Macro(
\ast 
)
] , \linebreak [0] 
[++Macro(
\ast 
)
] , \linebreak [0] 
[<<Macro(
\ast 
)
] , \linebreak [0] 
[||Macro(
\ast 
)
] , \linebreak [0] 
[01//Macro(
\ast 
)
] , \linebreak [0] 
[UB(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[LUB(
\ast 
, 
\ast 
)
] , \linebreak [0] 
[BS(
\ast 
,
\ast 
)
] , \linebreak [0] 
[UStelescope(
\ast 
,
\ast 
)
] , \linebreak [0] 
[(
\ast 
)
] , \linebreak [0] 
[|f
\ast 
|
] , \linebreak [0] 
[|r
\ast 
|
] , \linebreak [0] 
[Limit(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Union(
\ast 
)
] , \linebreak [0] 
[IsOrderedPair(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsRelation(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[isFunction(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsSeries(
\ast 
,
\ast 
)
] , \linebreak [0] 
[IsNatural(
\ast 
,\ast
)
] , \linebreak [0] 
[OrderedPair(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TypeNat(
\ast 
)
] , \linebreak [0] 
[TypeNat0(
\ast 
)
] , \linebreak [0] 
[TypeRational(
\ast 
)
] , \linebreak [0] 
[TypeRational0(
\ast 
)
] , \linebreak [0] 
[TypeSeries(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Typeseries0(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[\{
\ast 
,
\ast 
\}
] , \linebreak [0] 
[\langle 
\ast 
,
\ast 
\rangle
] , \linebreak [0] 
[(-u
\ast 
)
] , \linebreak [0] 
[-_{f}
\ast 
] , \linebreak [0] 
[(--
\ast 
)
] , \linebreak [0] 
[1f/
\ast 
] , \linebreak [0] 
[01//temp
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
(
\ast 
,
\ast 
)
] , \linebreak [0] 
[ReflRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[SymRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[TransRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[EqRel(
\ast 
,
\ast 
)
] , \linebreak [0] 
[[
\ast 
 \mathrel{\in} 
\ast 
]_{
\ast 
}
] , \linebreak [0] 
[Partition(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cdot 
\ast 
] , \linebreak [0] 
[
\ast 
\cdot_0 
\ast 
] , \linebreak [0] 
[(
\ast 
*
\ast 
)
] , \linebreak [0] 
[
\ast 
*_{f}
\ast 
] , \linebreak [0] 
[
\ast 
**
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
+ 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{+_1} 
\ast 
] , \linebreak [0] 
[
\ast 
- 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_0} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathop{-_1} 
\ast 
] , \linebreak [0] 
[(
\ast 
+
\ast 
)
] , \linebreak [0] 
[(
\ast 
-
\ast 
)
] , \linebreak [0] 
[
\ast 
+_{f}
\ast 
] , \linebreak [0] 
[
\ast 
-_{f}
\ast 
] , \linebreak [0] 
[
\ast 
++
\ast 
] , \linebreak [0] 
[R(
\ast 
) -- R(
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 \mathrel{\in} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[|
\ast 
|
] , \linebreak [0] 
[if(
\ast 
,
\ast 
,
\ast 
)
] , \linebreak [0] 
[Max(
\ast 
,
\ast 
)
] , \linebreak [0] 
[Max(
\ast 
,
\ast 
)
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
 = 
\ast 
] , \linebreak [0] 
[
\ast 
 \neq 
\ast 
] , \linebreak [0] 
[
\ast 
 <= 
\ast 
] , \linebreak [0] 
[
\ast 
 < 
\ast 
] , \linebreak [0] 
[
\ast 
<_{f}
\ast 
] , \linebreak [0] 
[
\ast 
\leq_{f}
\ast 
] , \linebreak [0] 
[SF(
\ast 
,
\ast 
)
] , \linebreak [0] 
[
\ast 
 == 
\ast 
] , \linebreak [0] 
[
\ast 
 !!== 
\ast 
] , \linebreak [0] 
[
\ast 
 << 
\ast 
] , \linebreak [0] 
[
\ast 
 <<== 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\cup \{ 
\ast 
\}
] , \linebreak [0] 
[
\ast 
\cup 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \{ 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel { \dot { . \, . } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { \dot { . \, . } } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { : \, : } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { \underline { {+} 2 \ast } }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { : \, : }
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel { {+} 2 \ast }
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
,\linebreak [0] 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\stackrel {B}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {D}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {C}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {P}{\approx }
\ast 
] , \linebreak [0] 
[
\ast 
\approx 
\ast 
] , \linebreak [0] 
[
\ast 
=
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {+}{\rightarrow }
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel {t^*}{=}
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{r}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_t 
\ast 
] , \linebreak [0] 
[
\ast 
\subseteq_T 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{T}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\stackrel{s}{=} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ in}^{\ast} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {free\ for}^{\ast} 
\ast 
\mathrel {in} 
\ast 
] , \linebreak [0] 
[
\ast 
\in_c 
\ast 
] , \linebreak [0] 
[
\ast 
< 
\ast 
] , \linebreak [0] 
[
\ast 
<' 
\ast 
] , \linebreak [0] 
[
\ast 
\le' 
\ast 
] , \linebreak [0] 
[
\ast 
= 
\ast 
] , \linebreak [0] 
[
\ast 
\neq 
\ast 
] , \linebreak [0] 
[
\ast 
{}^{var}
] , \linebreak [0] 
[
\ast 
\#^0
\ast 
] , \linebreak [0] 
[
\ast 
\#^1
\ast 
] , \linebreak [0] 
[
\ast 
\#^*
\ast 
] , \linebreak [0] 
[
\ast 
\!\mathrel{==}\! 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\subseteq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
{\neg }
\ast 
] , \linebreak [0] 
[\dot{\neg}\,(
\ast 
)n
] , \linebreak [0] 
[
\ast 
 \mathrel{\notin} 
\ast 
] , \linebreak [0] 
[
\ast 
 \mathrel{\neq} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\wedge 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\tilde{\wedge}} 
\ast 
] , \linebreak [0] 
[
\ast 
\wedge_c 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\wedge}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\vee 
\ast 
] , \linebreak [0] 
[
\ast 
\parallel 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\ddot{\vee}} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
 \mathrel{\dot{\vee}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\exists 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\forall_{obj} 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\exists 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\ddot{\Rightarrow}} 
\ast 
] , \linebreak [0] 
[
\ast 
\Rightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\Leftrightarrow 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{\dot{\Leftrightarrow}} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[ \{ ph \mathrel{\in} 
\ast 
\mid 
\ast 
\}
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
:
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{spy}
\ast 
] , \linebreak [0] 
[
\ast 
!
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\left\{\protect \begin {array}{l}
\ast 
\\
\ast 
\protect \end {array}\right.
]
; 
\newline \mathbf {Preassociative} \newline 
[
\lambda 
\ast 
.
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
. 
\ast 
] , \linebreak [0] 
[
\Lambda 
\ast 
] , \linebreak [0] 
[
{\bf if} \ 
\ast 
\ {\bf then} \ 
\ast 
\ {\bf else} \ 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
= 
\ast 
\mathbf{\ in\ } 
\ast 
] , \linebreak [0] 
[
\mathbf{let\ } 
\ast 
\mathrel{\ddot{=}} 
\ast 
\mathrel{\ in\ } 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\#
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
{} ^ { I }
] , \linebreak [0] 
[
\ast 
{} ^ { \rhd }
] , \linebreak [0] 
[
\ast 
{} ^ { V } 
] , \linebreak [0] 
[
\ast 
{} ^ { + } 
] , \linebreak [0] 
[
\ast 
{} ^ { - } 
] , \linebreak [0] 
[
\ast 
{} ^ { \ast } 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathop {\char64} 
\ast 
] , \linebreak [0] 
[
\ast 
\rhd 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} 
\ast 
] , \linebreak [0] 
[
\ast 
\gg 
\ast 
] , \linebreak [0] 
[
\ast 
\unrhd 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\vdash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel {i.e.} 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\forall 
\ast 
\colon 
\ast 
] , \linebreak [0] 
[
\Pi 
\ast 
\colon 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel {\oplus} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
; 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\ proves\ 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\mathbf{\ proof\ of\ } 
\ast 
: 
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: 
\ast 
\gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Last\ line \, 
\ast 
\gg 
\ast 
\,\Box
] , \linebreak [0] 
[
Line \, 
\ast 
: Premise \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Line \, 
\ast 
: \mbox{Side-condition} \gg 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
] , \linebreak [0] 
[
Local \gg 
\ast 
= 
\ast 
; 
\ast 
] , \linebreak [0] 
[
Begin \, 
\ast 
 ; 
\ast 
 : End ; 
\ast 
] , \linebreak [0] 
[
Last\ block\ line \, 
\ast 
\gg 
\ast 
\,;
] , \linebreak [0] 
[
Arbitrary \gg 
\ast 
;
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{|} 
\ast 
]
; 
\newline\mathbf{Postassociative} \newline 
[
\ast 
\mathrel{\mathrm{,}}
\ast 
] , \linebreak [0] 
[
\ast 
{[}
\ast 
{]}
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\& 
\ast 
]
; 
\newline \mathbf {Preassociative} \newline 
[
\ast 
\backslash \backslash 
\ast 
] , \linebreak [0] 
[
\ast 
\mathrel{linebreak[4]} 
\ast 
] , \linebreak [0] 
[
\ast 
\backslash \backslash 
\ast 
]
; 
]$

$\rm
[sup2
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}sup2
\mbox{''}
]$

$\rm
[sup2
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}sup2
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:1}{}\section* {$\rm LeqTotality(R)$}

$\rm
[LeqTotality(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash ToLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LeqTotality(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[LeqTotality(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqTotality(R)
\mbox{''}
]$

$\rm
[LeqTotality(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqTotality(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:2}{}\section* {$\rm PositiveToLeft(Eq)$}

$\rm
[PositiveToLeft(Eq)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
\vdash eqAddition
\rhd \underline{
\mathsf{x}}
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
\gg (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{z}}
)
)
 = ((\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
+(-u\underline{
\mathsf{z}}
)
)
; x=x+y-y
\gg \underline{
\mathsf{y}}
 = ((\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
+(-u\underline{
\mathsf{z}}
)
)
; eqSymmetry
\rhd \underline{
\mathsf{y}}
 = ((\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
+(-u\underline{
\mathsf{z}}
)
)
\gg ((\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
+(-u\underline{
\mathsf{z}}
)
)
 = \underline{
\mathsf{y}}
; eqTransitivity
\rhd (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{z}}
)
)
 = ((\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
+(-u\underline{
\mathsf{z}}
)
)
\rhd ((\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
+(-u\underline{
\mathsf{z}}
)
)
 = \underline{
\mathsf{y}}
\gg (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{z}}
)
)
 = \underline{
\mathsf{y}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveToLeft(Eq)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 = (\underline{
\mathsf{y}}
+\underline{
\mathsf{z}}
)
\vdash (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{z}}
)
)
 = \underline{
\mathsf{y}}
]$

$\rm
[PositiveToLeft(Eq)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToLeft(Eq)
\mbox{''}
]$

$\rm
[PositiveToLeft(Eq)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToLeft(Eq)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:3}{}\section* {$\rm ExpZero(Exact) $}

$\rm
[ExpZero(Exact) 
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon eqReflexivity
\gg 0
 = 0
; ExpZero
\rhd 0
 = 0
\gg  \underline{
\mathsf{x}}
 (exp) 0
 = 1
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[ExpZero(Exact) 
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon  \underline{
\mathsf{x}}
 (exp) 0
 = 1
]$

$\rm
[ExpZero(Exact) 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpZero(Exact)
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[ExpZero(Exact) 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expZero
\linebreak [0]\ \hskip0em plus2.0em{}exact
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:4}{}\section* {$\rm SameExp(Base)$}

$\rm
[SameExp(Base)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 0
 = \underline{
\mathsf {n}}
\vdash ExpZero(Exact) 
\gg  \underline{
\mathsf{x}}
 (exp) 0
 = 1
; eqSymmetry
\rhd 0
 = \underline{
\mathsf {n}}
\gg \underline{
\mathsf {n}}
 = 0
; ExpZero
\rhd \underline{
\mathsf {n}}
 = 0
\gg  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
 = 1
; eqSymmetry
\rhd  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
 = 1
\gg 1
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; eqTransitivity
\rhd  \underline{
\mathsf{x}}
 (exp) 0
 = 1
\rhd 1
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 0
 = \underline{
\mathsf {n}}
\vdash  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg 0
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; Gen
\rhd 0
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon 0
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameExp(Base)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall_{obj} \underline{
\mathsf {n}}
\colon 0
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
]$

$\rm
[SameExp(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameExp(Base)
\mbox{''}
]$

$\rm
[SameExp(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:5}{}\section* {$\rm SameExp(Indu)$}

$\rm
[SameExp(Indu)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\vdash (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\vdash (+1)IsPositive(N)
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
; ExpPositive
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\gg  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
; x=x+y-y
\gg \underline{
\mathsf {m}}
 = ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
; A4
\mathop {\char64} ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg \underline{
\mathsf {m}}
 = ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
; MP
\rhd \underline{
\mathsf {m}}
 = ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\rhd \underline{
\mathsf {m}}
 = ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\gg  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
; eqSymmetry
\rhd  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\gg  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
; EqMultiplicationLeft
\rhd  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\gg (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
; A4
\mathop {\char64} (\underline{
\mathsf {n}}
+(-u1
)
)
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg \underline{
\mathsf {m}}
 = (\underline{
\mathsf {n}}
+(-u1
)
)
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
; PositiveToRight(Eq)
\rhd (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\gg \underline{
\mathsf {m}}
 = (\underline{
\mathsf {n}}
+(-u1
)
)
; MP
\rhd \underline{
\mathsf {m}}
 = (\underline{
\mathsf {n}}
+(-u1
)
)
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
\rhd \underline{
\mathsf {m}}
 = (\underline{
\mathsf {n}}
+(-u1
)
)
\gg  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
; EqMultiplicationLeft
\rhd  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
\gg (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
)
; SubLessRight
\rhd (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {n}}
)n
)n
)n
; ExpPositive
\rhd \dot{\neg}\,(0
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf {n}}
)n
)n
)n
\gg  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
)
; eqSymmetry
\rhd  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
)
\gg (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; eqTransitivity5
\rhd  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
\rhd (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
\rhd (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
)
\rhd (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {n}}
+(-u1
)
)
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\vdash (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\vdash  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\Rightarrow (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\vdash MP
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\Rightarrow (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; Gen
\rhd (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\vdash 
\forall_{obj} \underline{
\mathsf {n}}
\colon (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\Rightarrow 
\forall_{obj} \underline{
\mathsf {n}}
\colon (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameExp(Indu)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\Rightarrow 
\forall_{obj} \underline{
\mathsf {n}}
\colon (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
]$

$\rm
[SameExp(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameExp(Indu)
\mbox{''}
]$

$\rm
[SameExp(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:6}{}\section* {$\rm SameExp$}

$\rm
[SameExp
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\vdash SameExp(Base)
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon 0
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; SameExp(Indu)
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\Rightarrow 
\forall_{obj} \underline{
\mathsf {n}}
\colon (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; Induction
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon 0
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) 0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\Rightarrow 
\forall_{obj} \underline{
\mathsf {n}}
\colon (\underline{
\mathsf {m}}
+1
)
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; A4
\mathop {\char64} \underline{
\mathsf {n}}
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\gg \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
; MP
\rhd \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\Rightarrow  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\rhd \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\gg  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameExp
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \underline{
\mathsf {m}}
 = \underline{
\mathsf {n}}
\vdash  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {n}}
]$

$\rm
[SameExp
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameExp
\mbox{''}
]$

$\rm
[SameExp
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:7}{}\section* {$\rm (1/2)(x+y)-x=(1/2)(y-x)$}

$\rm
[(1/2)(x+y)-x=(1/2)(y-x)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon Distribution
\gg (rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
; eqAddition
\rhd (rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
\gg ((rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = (((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
; plusCommutativity
\gg ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
; eqAddition
\rhd ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
\gg (((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = (((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
; plusAssociativity
\gg (((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
)
; TwoHalves
\gg ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
 = \underline{
\mathsf{x}}
; PositiveToRight(Eq)
\rhd ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
 = \underline{
\mathsf{x}}
\gg (rec(1
+1
)
*\underline{
\mathsf{x}}
)
 = (\underline{
\mathsf{x}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; EqNegated
\rhd (rec(1
+1
)
*\underline{
\mathsf{x}}
)
 = (\underline{
\mathsf{x}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\gg (-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
 = (-u(\underline{
\mathsf{x}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
; MinusNegated
\gg (-u(\underline{
\mathsf{x}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
; eqTransitivity
\rhd (-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
 = (-u(\underline{
\mathsf{x}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
\rhd (-u(\underline{
\mathsf{x}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
\gg (-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
; eqSymmetry
\rhd (-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
\gg ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
 = (-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
; EqAdditionLeft
\rhd ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
 = (-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
\gg ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; DistributionOut(Minus)
\gg ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
; eqTransitivity6
\rhd ((rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = (((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
\rhd (((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = (((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
\rhd (((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
)
\rhd ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(-u\underline{
\mathsf{x}}
)
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\rhd ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
\gg ((rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[(1/2)(x+y)-x=(1/2)(y-x)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon ((rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
+(-u\underline{
\mathsf{x}}
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
]$

$\rm
[(1/2)(x+y)-x=(1/2)(y-x)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]$

$\rm
[(1/2)(x+y)-x=(1/2)(y-x)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:8}{}\section* {$\rm y-(1/2)(x+y)=(1/2)(y-x)$}

$\rm
[y-(1/2)(x+y)=(1/2)(y-x)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon Distribution
\gg (rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
; EqNegated
\rhd (rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
\gg (-u(rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
)
 = (-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
; EqAdditionLeft
\rhd (-u(rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
)
 = (-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
\gg (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
)
)
 = (\underline{
\mathsf{y}}
+(-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
)
; plusCommutativity
\gg ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
; EqNegated
\rhd ((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
\gg (-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
 = (-u((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; -x-y=-(x+y)
\gg ((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (-u((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; eqSymmetry
\rhd ((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (-u((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\gg (-u((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = ((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; eqTransitivity
\rhd (-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
 = (-u((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\rhd (-u((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = ((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\gg (-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
 = ((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; EqAdditionLeft
\rhd (-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
 = ((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\gg (\underline{
\mathsf{y}}
+(-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
)
 = (\underline{
\mathsf{y}}
+((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
; plusAssociativity
\gg ((\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (\underline{
\mathsf{y}}
+((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
; eqSymmetry
\rhd ((\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (\underline{
\mathsf{y}}
+((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
\gg (\underline{
\mathsf{y}}
+((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
 = ((\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; TwoHalves
\gg ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
 = \underline{
\mathsf{y}}
; PositiveToRight(Eq)
\rhd ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
 = \underline{
\mathsf{y}}
\gg (rec(1
+1
)
*\underline{
\mathsf{y}}
)
 = (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
; eqSymmetry
\rhd (rec(1
+1
)
*\underline{
\mathsf{y}}
)
 = (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
\gg (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
 = (rec(1
+1
)
*\underline{
\mathsf{y}}
)
; eqAddition
\rhd (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
 = (rec(1
+1
)
*\underline{
\mathsf{y}}
)
\gg ((\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
; DistributionOut(Minus)
\gg ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
; eqTransitivity6
\rhd (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
)
)
 = (\underline{
\mathsf{y}}
+(-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
)
\rhd (\underline{
\mathsf{y}}
+(-u((rec(1
+1
)
*\underline{
\mathsf{x}}
)
+(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
)
 = (\underline{
\mathsf{y}}
+((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
\rhd (\underline{
\mathsf{y}}
+((-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
)
 = ((\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\rhd ((\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*\underline{
\mathsf{y}}
)
)
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
\rhd ((rec(1
+1
)
*\underline{
\mathsf{y}}
)
+(-u(rec(1
+1
)
*\underline{
\mathsf{x}}
)
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
\gg (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[y-(1/2)(x+y)=(1/2)(y-x)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon (\underline{
\mathsf{y}}
+(-u(rec(1
+1
)
*(\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
)
)
)
 = (rec(1
+1
)
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)
]$

$\rm
[y-(1/2)(x+y)=(1/2)(y-x)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]$

$\rm
[y-(1/2)(x+y)=(1/2)(y-x)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:9}{}\section* {$\rm BSzero(Exact)$}

$\rm
[BSzero(Exact)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon eqReflexivity
\gg 0
 = 0
; BSzero
\rhd 0
 = 0
\gg BS(\underline{
\mathsf {m}}
,0
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[BSzero(Exact)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon BS(\underline{
\mathsf {m}}
,0
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
]$

$\rm
[BSzero(Exact)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSzero(Exact)
\mbox{''}
]$

$\rm
[BSzero(Exact)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}exact
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:10}{}\section* {$\rm SameBS(2)(Base)$}

$\rm
[SameBS(2)(Base)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon 0
 = \underline{(n2)}
\vdash eqSymmetry
\rhd 0
 = \underline{(n2)}
\gg \underline{(n2)}
 = 0
; BSzero
\rhd \underline{(n2)}
 = 0
\gg BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; eqSymmetry
\rhd BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; BSzero(Exact)
\gg BS(\underline{
\mathsf {m}}
,0
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; eqTransitivity
\rhd BS(\underline{
\mathsf {m}}
,0
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\rhd  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon 0
 = \underline{(n2)}
\vdash BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 0
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Gen
\rhd 0
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameBS(2)(Base)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
]$

$\rm
[SameBS(2)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameBS(2)(Base)
\mbox{''}
]$

$\rm
[SameBS(2)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameBase(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:11}{}\section* {$\rm SameBS(2)(Indu)$}

$\rm
[SameBS(2)(Indu)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash (\underline{(n1)}
+1
)
 = \underline{(n2)}
\vdash (+1)IsPositive(N)
\gg \dot{\neg}\,(0
 <= (\underline{(n1)}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{(n1)}
+1
)
)n
)n
)n
; BSpositive
\rhd \dot{\neg}\,(0
 <= (\underline{(n1)}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{(n1)}
+1
)
)n
)n
)n
\gg BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
; x=x+y-y
\gg \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
; A4
\mathop {\char64} ((\underline{(n1)}
+1
)
+(-u1
)
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
; MP
\rhd \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
\rhd \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
\gg BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
; eqSymmetry
\rhd BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
\gg BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
; EqAdditionLeft
\rhd BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
\gg ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
)
; PositiveToRight(Eq)
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\gg \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
; A4
\mathop {\char64} (\underline{(n2)}
+(-u1
)
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
; MP
\rhd \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
\rhd \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
\gg BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
; EqAdditionLeft
\rhd BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
\gg ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
; EqAdditionLeft
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\gg (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
 = (\underline{
\mathsf {m}}
+\underline{(n2)}
)
; SameExp
\rhd (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
 = (\underline{
\mathsf {m}}
+\underline{(n2)}
)
\gg  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
; eqAddition
\rhd  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
\gg ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
; SubLessRight
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\rhd \dot{\neg}\,(0
 <= (\underline{(n1)}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{(n1)}
+1
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <= \underline{(n2)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(n2)}
)n
)n
)n
; BSpositive
\rhd \dot{\neg}\,(0
 <= \underline{(n2)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(n2)}
)n
)n
)n
\gg BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
; eqSymmetry
\rhd BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
\gg ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; eqTransitivity6
\rhd BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
\rhd ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
)
\rhd ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
\rhd ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
\rhd ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+\underline{(n2)}
)
+BS(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash (\underline{(n1)}
+1
)
 = \underline{(n2)}
\vdash BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash MP
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Gen
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameBS(2)(Indu)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
]$

$\rm
[SameBS(2)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameBS(2)(Indu)
\mbox{''}
]$

$\rm
[SameBS(2)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameBase(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:12}{}\section* {$\rm SameBS(2)$}

$\rm
[SameBS(2)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\vdash SameBS(2)(Base)
\gg 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; SameBS(2)(Indu)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Induction
\rhd 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,0
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; A4
\mathop {\char64} \underline{(n2)}
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; MP
\rhd \underline{(n1)}
 = \underline{(n2)}
\Rightarrow BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rhd \underline{(n1)}
 = \underline{(n2)}
\gg BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameBS(2)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\vdash BS(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = BS(\underline{
\mathsf {m}}
,\underline{(n2)}
)
]$

$\rm
[SameBS(2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameBS(2)
\mbox{''}
]$

$\rm
[SameBS(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameBase(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:13}{}\section* {$\rm NegativeToLeft(Less)(1 term)$}

$\rm
[NegativeToLeft(Less)(1 term)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(0
 <= (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
)n
)n
)n
\vdash LessAddition
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
)n
)n
)n
\gg \dot{\neg}\,((0
+\underline{
\mathsf{y}}
)
 <= ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,((0
+\underline{
\mathsf{y}}
)
 = ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
)n
)n
)n
; plus0Left
\gg (0
+\underline{
\mathsf{y}}
)
 = \underline{
\mathsf{y}}
; SubLessLeft
\rhd (0
+\underline{
\mathsf{y}}
)
 = \underline{
\mathsf{y}}
\rhd \dot{\neg}\,((0
+\underline{
\mathsf{y}}
)
 <= ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,((0
+\underline{
\mathsf{y}}
)
 = ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf{y}}
 <= ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{y}}
 = ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
)n
)n
)n
; Three2threeTerms
\gg ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
 = ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+(-u\underline{
\mathsf{y}}
)
)
; x=x+y-y
\gg \underline{
\mathsf{x}}
 = ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+(-u\underline{
\mathsf{y}}
)
)
; eqSymmetry
\rhd \underline{
\mathsf{x}}
 = ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+(-u\underline{
\mathsf{y}}
)
)
\gg ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+(-u\underline{
\mathsf{y}}
)
)
 = \underline{
\mathsf{x}}
; eqTransitivity
\rhd ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
 = ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+(-u\underline{
\mathsf{y}}
)
)
\rhd ((\underline{
\mathsf{x}}
+\underline{
\mathsf{y}}
)
+(-u\underline{
\mathsf{y}}
)
)
 = \underline{
\mathsf{x}}
\gg ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
 = \underline{
\mathsf{x}}
; SubLessRight
\rhd ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
 = \underline{
\mathsf{x}}
\rhd \dot{\neg}\,(\underline{
\mathsf{y}}
 <= ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{y}}
 = ((\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
+\underline{
\mathsf{y}}
)
)n
)n
)n
\gg \dot{\neg}\,(\underline{
\mathsf{y}}
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{y}}
 = \underline{
\mathsf{x}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeToLeft(Less)(1 term)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(0
 <= (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{x}}
+(-u\underline{
\mathsf{y}}
)
)
)n
)n
)n
\vdash \dot{\neg}\,(\underline{
\mathsf{y}}
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{y}}
 = \underline{
\mathsf{x}}
)n
)n
)n
]$

$\rm
[NegativeToLeft(Less)(1 term)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$

$\rm
[NegativeToLeft(Less)(1 term)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:14}{}\section* {$\rm BS(+1)$}

$\rm
[BS(+1)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon (+1)IsPositive(N)
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf {n}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {n}}
+1
)
)n
)n
)n
; BSpositive
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf {n}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {n}}
+1
)
)n
)n
)n
\gg BS(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
; plusAssociativity
\gg ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
 = (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
; SameExp
\rhd ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
 = (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
\gg  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
; eqSymmetry
\rhd  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
\gg  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
; x=x+y-y
\gg \underline{
\mathsf {n}}
 = ((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
; SameBS(2)
\rhd \underline{
\mathsf {n}}
 = ((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
\gg BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
; eqSymmetry
\rhd BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
 = BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
\gg BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
; AddEquations
\rhd  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
\rhd BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
 = BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\gg ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
 = ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
+BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
; eqTransitivity
\rhd BS(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
\rhd ( rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
+BS(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
 = ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
+BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
\gg BS(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
+BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[BS(+1)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon BS(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+\underline{
\mathsf {n}}
)
+1
)
+BS(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
]$

$\rm
[BS(+1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BS(
{+}1)
\mbox{''}
]$

$\rm
[BS(+1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum(
{+}1)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:15}{}\section* {$\rm BSbound(Exact)(Base)$}

$\rm
[BSbound(Exact)(Base)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon BSzero(Exact)
\gg BS((\underline{
\mathsf {m}}
+1
)
,0
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
; Exp(+1)
\gg  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
; eqTransitivity
\rhd BS((\underline{
\mathsf {m}}
+1
)
,0
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
\rhd  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
\gg BS((\underline{
\mathsf {m}}
+1
)
,0
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
; plus0
\gg ((\underline{
\mathsf {m}}
+1
)
+0
)
 = (\underline{
\mathsf {m}}
+1
)
; SameExp
\rhd ((\underline{
\mathsf {m}}
+1
)
+0
)
 = (\underline{
\mathsf {m}}
+1
)
\gg  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
; eqTransitivity
\rhd  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
 =  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
\rhd  rec(1
+1
)
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
\gg  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
; AddEquations
\rhd BS((\underline{
\mathsf {m}}
+1
)
,0
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
\rhd  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
\gg (BS((\underline{
\mathsf {m}}
+1
)
,0
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
)
 = ((rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
)
; TwoHalves
\gg ((rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; eqTransitivity
\rhd (BS((\underline{
\mathsf {m}}
+1
)
,0
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
)
 = ((rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
)
\rhd ((rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg (BS((\underline{
\mathsf {m}}
+1
)
,0
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[BSbound(Exact)(Base)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon (BS((\underline{
\mathsf {m}}
+1
)
,0
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
]$

$\rm
[BSbound(Exact)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSbound(Exact)(Base)
\mbox{''}
]$

$\rm
[BSbound(Exact)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}exact
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:16}{}\section* {$\rm BSbound(Exact)(Indu)$}

$\rm
[BSbound(Exact)(Indu)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\vdash BS(+1)
\gg BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
 = ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
; eqAddition
\rhd BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
 = ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
\gg (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = (( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
; plusCommutativity
\gg ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
 = (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
; eqAddition
\rhd ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
 = (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
\gg (( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = ((BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
; Exp(+1)
\gg  rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
; AddEquations
\rhd  rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
\rhd  rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
 = (rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
\gg ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = ((rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
)
; TwoHalves
\gg ((rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
)
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
; eqTransitivity
\rhd ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = ((rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
)
\rhd ((rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
+(rec(1
+1
)
* rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
)
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
\gg ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
; Three2twoTerms
\rhd ( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
\gg ((BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
; eqTransitivity5
\rhd (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = (( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
\rhd (( rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = ((BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
\rhd ((BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 = (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
\rhd (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\vdash (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\Rightarrow (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[BSbound(Exact)(Indu)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\Rightarrow (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
]$

$\rm
[BSbound(Exact)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSbound(Exact)(Indu)
\mbox{''}
]$

$\rm
[BSbound(Exact)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}exact
\linebreak [0]\ \hskip0em plus2.0em{}bound
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:17}{}\section* {$\rm BSbound(Exact)$}

$\rm
[BSbound(Exact)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon BSbound(Exact)(Base)
\gg (BS((\underline{
\mathsf {m}}
+1
)
,0
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; BSbound(Exact)(Indu)
\gg (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\Rightarrow (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; Induction
\rhd (BS((\underline{
\mathsf {m}}
+1
)
,0
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+0
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\rhd (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\Rightarrow (BS((\underline{
\mathsf {m}}
+1
)
,(\underline{
\mathsf {n}}
+1
)
)
+ rec(1
+1
)
 (exp) (((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+1
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[BSbound(Exact)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
]$

$\rm
[BSbound(Exact)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSbound(Exact)
\mbox{''}
]$

$\rm
[BSbound(Exact)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}exact
\linebreak [0]\ \hskip0em plus2.0em{}bound
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:18}{}\section* {$\rm BSbound$}

$\rm
[BSbound
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon BSbound(Exact)
\gg (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; plusCommutativity
\gg ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
 = (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
; eqTransitivity
\rhd ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
 = (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
\rhd (BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
+ rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
; PositiveToRight(Eq)
\rhd ( rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
+BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\gg  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
 = ( rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
+(-uBS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
)
; 0<1/2
\gg \dot{\neg}\,(0
 <= rec(1
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = rec(1
+1
)
)n
)n
)n
; PositiveBase
\rhd \dot{\neg}\,(0
 <= rec(1
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = rec(1
+1
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <=  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)n
)n
)n
; SubLessRight
\rhd  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
 = ( rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
+(-uBS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
)
\rhd \dot{\neg}\,(0
 <=  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  rec(1
+1
)
 (exp) ((\underline{
\mathsf {m}}
+1
)
+\underline{
\mathsf {n}}
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <= ( rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
+(-uBS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = ( rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
+(-uBS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
)
)n
)n
)n
; NegativeToLeft(Less)(1 term)
\rhd \dot{\neg}\,(0
 <= ( rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
+(-uBS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = ( rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
+(-uBS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
)
)
)n
)n
)n
\gg \dot{\neg}\,(BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
 <=  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[BSbound
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon \dot{\neg}\,(BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
 <=  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(BS((\underline{
\mathsf {m}}
+1
)
,\underline{
\mathsf {n}}
)
 =  rec(1
+1
)
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
]$

$\rm
[BSbound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}BSbound
\mbox{''}
]$

$\rm
[BSbound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}base(1/2)Sum
\linebreak [0]\ \hskip0em plus2.0em{}bound
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:19}{}\section* {$\rm UStelescope(Zero)(Exact)$}

$\rm
[UStelescope(Zero)(Exact)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon eqReflexivity
\gg 0
 = 0
; UStelescope(Zero)
\rhd 0
 = 0
\gg UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[UStelescope(Zero)(Exact)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
]$

$\rm
[UStelescope(Zero)(Exact)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UStelescope(Zero)(Exact)
\mbox{''}
]$

$\rm
[UStelescope(Zero)(Exact)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}UStelescope
\linebreak [0]\ \hskip0em plus2.0em{}zero
\linebreak [0]\ \hskip0em plus2.0em{}exact
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:20}{}\section* {$\rm SameTelescope(2)(Base)$}

$\rm
[SameTelescope(2)(Base)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon 0
 = \underline{(n2)}
\vdash eqSymmetry
\rhd 0
 = \underline{(n2)}
\gg \underline{(n2)}
 = 0
; UStelescope(Zero)
\rhd \underline{(n2)}
 = 0
\gg UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
; eqSymmetry
\rhd UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; UStelescope(Zero)(Exact)
\gg UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
; eqTransitivity
\rhd UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon 0
 = \underline{(n2)}
\vdash UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 0
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Gen
\rhd 0
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameTelescope(2)(Base)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
]$

$\rm
[SameTelescope(2)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameTelescope(2)(Base)
\mbox{''}
]$

$\rm
[SameTelescope(2)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameTelescope
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:21}{}\section* {$\rm SameTelescope(2)(Indu)$}

$\rm
[SameTelescope(2)(Indu)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash (\underline{(n1)}
+1
)
 = \underline{(n2)}
\vdash (+1)IsPositive(N)
\gg \dot{\neg}\,(0
 <= (\underline{(n1)}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{(n1)}
+1
)
)n
)n
)n
; UStelescope(Positive)
\rhd \dot{\neg}\,(0
 <= (\underline{(n1)}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{(n1)}
+1
)
)n
)n
)n
\gg UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
; x=x+y-y
\gg \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
; A4
\mathop {\char64} ((\underline{(n1)}
+1
)
+(-u1
)
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
; MP
\rhd \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
\rhd \underline{(n1)}
 = ((\underline{(n1)}
+1
)
+(-u1
)
)
\gg UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
; eqSymmetry
\rhd UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
\gg UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
; PositiveToRight(Eq)
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\gg \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
; A4
\mathop {\char64} (\underline{(n2)}
+(-u1
)
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
; MP
\rhd \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
\rhd \underline{(n1)}
 = (\underline{(n2)}
+(-u1
)
)
\gg UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
; eqTransitivity
\rhd UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
\rhd UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
\gg UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
; EqAdditionLeft
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\gg (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
 = (\underline{
\mathsf {m}}
+\underline{(n2)}
)
; eqAddition
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\gg ((\underline{(n1)}
+1
)
+1
)
 = (\underline{(n2)}
+1
)
; EqAdditionLeft
\rhd ((\underline{(n1)}
+1
)
+1
)
 = (\underline{(n2)}
+1
)
\gg (\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
 = (\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
; SameSeries(NumDiff)
\rhd (\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
 = (\underline{
\mathsf {m}}
+\underline{(n2)}
)
\rhd (\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
 = (\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
\gg |(us
[(\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
]
)
)
|
 = |(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
; AddEquations
\rhd |(us
[(\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
]
)
)
|
 = |(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
\rhd UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
\gg (|(us
[(\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
; SubLessRight
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\rhd \dot{\neg}\,(0
 <= (\underline{(n1)}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{(n1)}
+1
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <= \underline{(n2)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(n2)}
)n
)n
)n
; UStelescope(Positive)
\rhd \dot{\neg}\,(0
 <= \underline{(n2)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(n2)}
)n
)n
)n
\gg UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 = (|(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
; eqSymmetry
\rhd UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
 = (|(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
\gg (|(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; eqTransitivity4
\rhd UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
\rhd (|(us
[(\underline{
\mathsf {m}}
+(\underline{(n1)}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{(n1)}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{(n1)}
+1
)
+(-u1
)
)
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
\rhd (|(us
[(\underline{
\mathsf {m}}
+\underline{(n2)}
)
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{(n2)}
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,(\underline{(n2)}
+(-u1
)
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash (\underline{(n1)}
+1
)
 = \underline{(n2)}
\vdash UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash MP
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Gen
\rhd (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\vdash 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameTelescope(2)(Indu)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
]$

$\rm
[SameTelescope(2)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameTelescope(2)(Indu)
\mbox{''}
]$

$\rm
[SameTelescope(2)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameTelescope
\linebreak [0]\ \hskip0em plus2.0em{}second
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:22}{}\section* {$\rm SameTelescope(2)$}

$\rm
[SameTelescope(2)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\vdash SameTelescope(2)(Base)
\gg 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; SameTelescope(2)(Indu)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; Induction
\rhd 
\forall_{obj} \underline{(n2)}
\colon 0
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,0
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\Rightarrow 
\forall_{obj} \underline{(n2)}
\colon (\underline{(n1)}
+1
)
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,(\underline{(n1)}
+1
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; A4
\mathop {\char64} \underline{(n2)}
\rhd 
\forall_{obj} \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\gg \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
; MP
\rhd \underline{(n1)}
 = \underline{(n2)}
\Rightarrow UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rhd \underline{(n1)}
 = \underline{(n2)}
\gg UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameTelescope(2)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon \underline{(n1)}
 = \underline{(n2)}
\vdash UStelescope(\underline{
\mathsf {m}}
,\underline{(n1)}
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{(n2)}
)
]$

$\rm
[SameTelescope(2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameTelescope(2)
\mbox{''}
]$

$\rm
[SameTelescope(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameTelescope
\linebreak [0]\ \hskip0em plus2.0em{}second
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:23}{}\section* {$\rm Exp(+1)$}

$\rm
[Exp(+1)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon (+1)IsPositive(N)
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
; ExpPositive
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\gg  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
; x=x+y-y
\gg \underline{
\mathsf {m}}
 = ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
; SameExp
\rhd \underline{
\mathsf {m}}
 = ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\gg  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
; EqMultiplicationLeft
\rhd  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
 =  \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
\gg (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
; eqSymmetry
\rhd (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
\gg (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
; eqTransitivity
\rhd  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
\rhd (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) ((\underline{
\mathsf {m}}
+1
)
+(-u1
)
)
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
\gg  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[Exp(+1)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
]$

$\rm
[Exp(+1)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Exp(
{+}1)
\mbox{''}
]$

$\rm
[Exp(+1)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}exp(
{+}1)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:24}{}\section* {$\rm PositiveBase(Base)$}

$\rm
[PositiveBase(Base)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon ExpZero(Exact) 
\gg  \underline{
\mathsf{x}}
 (exp) 0
 = 1
; eqSymmetry
\rhd  \underline{
\mathsf{x}}
 (exp) 0
 = 1
\gg 1
 =  \underline{
\mathsf{x}}
 (exp) 0
; 0<1
\gg \dot{\neg}\,(0
 <= 1
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = 1
)n
)n
)n
; SubLessRight
\rhd 1
 =  \underline{
\mathsf{x}}
 (exp) 0
\rhd \dot{\neg}\,(0
 <= 1
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = 1
)n
)n
)n
\gg \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) 0
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) 0
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveBase(Base)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) 0
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) 0
)n
)n
)n
]$

$\rm
[PositiveBase(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveBase(Base)
\mbox{''}
]$

$\rm
[PositiveBase(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:25}{}\section* {$\rm PositiveBase(Indu)$}

$\rm
[PositiveBase(Indu)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\vdash \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\vdash Exp(+1)
\gg  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
; eqSymmetry
\rhd  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
\gg (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
; PositiveFactors
\rhd \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\rhd \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
)n
)n
)n
; SubLessRight
\rhd (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{x}}
* \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\vdash \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\vdash \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
; \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\vdash MP
\rhd \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\rhd \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\gg \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveBase(Indu)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\vdash \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
]$

$\rm
[PositiveBase(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveBase(Indu)
\mbox{''}
]$

$\rm
[PositiveBase(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:26}{}\section* {$\rm PositiveBase$}

$\rm
[PositiveBase
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\vdash PositiveBase(Base)
\gg \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) 0
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) 0
)n
)n
)n
; PositiveBase(Indu)
\rhd \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\gg \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
; Induction
\rhd \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) 0
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) 0
)n
)n
)n
\rhd \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\gg \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveBase
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf{x}}
\colon \dot{\neg}\,(0
 <= \underline{
\mathsf{x}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{
\mathsf{x}}
)n
)n
)n
\vdash \dot{\neg}\,(0
 <=  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 =  \underline{
\mathsf{x}}
 (exp) \underline{
\mathsf {m}}
)n
)n
)n
]$

$\rm
[PositiveBase
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveBase
\mbox{''}
]$

$\rm
[PositiveBase
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:27}{}\section* {$\rm TelescopeNumerical(Base)$}

$\rm
[TelescopeNumerical(Base)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon eqReflexivity
\gg 0
 = 0
; UStelescope(Zero)
\rhd 0
 = 0
\gg UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
; eqReflexivity
\gg \underline{
\mathsf {m}}
 = \underline{
\mathsf {m}}
; plus0Left
\gg (0
+1
)
 = 1
; EqAdditionLeft
\rhd (0
+1
)
 = 1
\gg (\underline{
\mathsf {m}}
+(0
+1
)
)
 = (\underline{
\mathsf {m}}
+1
)
; eqSymmetry
\rhd (\underline{
\mathsf {m}}
+(0
+1
)
)
 = (\underline{
\mathsf {m}}
+1
)
\gg (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf {m}}
+(0
+1
)
)
; SameSeries(NumDiff)
\rhd \underline{
\mathsf {m}}
 = \underline{
\mathsf {m}}
\rhd (\underline{
\mathsf {m}}
+1
)
 = (\underline{
\mathsf {m}}
+(0
+1
)
)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
; eqTransitivity
\rhd UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+1
)
]
)
)
|
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
\gg UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
; eqSymmetry
\rhd UStelescope(\underline{
\mathsf {m}}
,0
)
 = |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
 = UStelescope(\underline{
\mathsf {m}}
,0
)
; eqLeq
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
 = UStelescope(\underline{
\mathsf {m}}
,0
)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,0
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[TelescopeNumerical(Base)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,0
)
]$

$\rm
[TelescopeNumerical(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TelescopeNumerical(Base)
\mbox{''}
]$

$\rm
[TelescopeNumerical(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}telescopeNumerical
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:28}{}\section* {$\rm TelescopeNumerical(Indu)$}

$\rm
[TelescopeNumerical(Indu)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\vdash (+1)IsPositive(N)
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf {n}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {n}}
+1
)
)n
)n
)n
; UStelescope(Positive)
\rhd \dot{\neg}\,(0
 <= (\underline{
\mathsf {n}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {n}}
+1
)
)n
)n
)n
\gg UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
; x=x+y-y
\gg \underline{
\mathsf {n}}
 = ((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
; eqSymmetry
\rhd \underline{
\mathsf {n}}
 = ((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
\gg ((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
 = \underline{
\mathsf {n}}
; SameTelescope(2)
\rhd ((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
 = \underline{
\mathsf {n}}
\gg UStelescope(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
; EqAdditionLeft
\rhd UStelescope(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
 = UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\gg (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
; eqTransitivity
\rhd UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
\rhd (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,((\underline{
\mathsf {n}}
+1
)
+(-u1
)
)
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
\gg UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
; eqSymmetry
\rhd UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
 = (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
\gg (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
; LeqAdditionLeft
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\gg (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
)
 <= (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
; subLeqRight
\rhd (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
 = UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
\rhd (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
)
 <= (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
)
\gg (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
)
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
; plusCommutativity
\gg (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
)
 = (|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
)
; subLeqLeft
\rhd (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
)
 = (|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
)
\rhd (|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
+|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
)
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
\gg (|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
)
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
; insertMiddleTerm(Numerical)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= (|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
)
; leqTransitivity
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= (|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
)
\rhd (|(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
+|(us
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
)
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\vdash |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\Rightarrow |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[TelescopeNumerical(Indu)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\Rightarrow |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
]$

$\rm
[TelescopeNumerical(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TelescopeNumerical(Indu)
\mbox{''}
]$

$\rm
[TelescopeNumerical(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}telescopeNumerical
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:29}{}\section* {$\rm TelescopeNumerical$}

$\rm
[TelescopeNumerical
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon TelescopeNumerical(Base)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,0
)
; TelescopeNumerical(Indu)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\Rightarrow |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
; Induction
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(0
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,0
)
\rhd |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\Rightarrow |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+((\underline{
\mathsf {n}}
+1
)
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,(\underline{
\mathsf {n}}
+1
)
)
\gg |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[TelescopeNumerical
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon |(us
[\underline{
\mathsf {m}}
]
+(-uus
[(\underline{
\mathsf {m}}
+(\underline{
\mathsf {n}}
+1
)
)
]
)
)
|
 <= UStelescope(\underline{
\mathsf {m}}
,\underline{
\mathsf {n}}
)
]$

$\rm
[TelescopeNumerical
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}TelescopeNumerical
\mbox{''}
]$

$\rm
[TelescopeNumerical
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}telescopeNumerical
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:30}{}\section* {$\rm (+1)IsPositive(N)$}

$\rm
[(+1)IsPositive(N)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon Nat(\underline{
\mathsf {m}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} Nonnegative(N)
\mathrel {\makebox [0mm][l]{$\rhd $}\,{\rhd }} Nat(\underline{
\mathsf {m}}
)
\gg 0
 <= \underline{
\mathsf {m}}
; Leq+1
\rhd 0
 <= \underline{
\mathsf {m}}
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[(+1)IsPositive(N)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon Nat(\underline{
\mathsf {m}}
)
\mathrel {\makebox [0mm][l]{$\vdash $}\,{\vdash }} \dot{\neg}\,(0
 <= (\underline{
\mathsf {m}}
+1
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf {m}}
+1
)
)n
)n
)n
]$

$\rm
[(+1)IsPositive(N)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
{+}1)IsPositive(N)
\mbox{''}
]$

$\rm
[(+1)IsPositive(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
{+}1IsPositive(N)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:31}{}\section* {$\rm DistributionOut(Minus)$}

$\rm
[DistributionOut(Minus)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon Times(-1)Left
\gg ((-u1
)
*(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = (-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
; eqSymmetry
\rhd ((-u1
)
*(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = (-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
\gg (-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = ((-u1
)
*(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
; timesCommutativity
\gg ((-u1
)
*(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = ((\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
*(-u1
)
)
; timesAssociativity
\gg ((\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
*(-u1
)
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{z}}
*(-u1
)
)
)
; Times(-1)
\gg (\underline{
\mathsf{z}}
*(-u1
)
)
 = (-u\underline{
\mathsf{z}}
)
; EqMultiplicationLeft
\rhd (\underline{
\mathsf{z}}
*(-u1
)
)
 = (-u\underline{
\mathsf{z}}
)
\gg (\underline{
\mathsf{x}}
*(\underline{
\mathsf{z}}
*(-u1
)
)
)
 = (\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
; eqTransitivity5
\rhd (-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = ((-u1
)
*(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
\rhd ((-u1
)
*(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = ((\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
*(-u1
)
)
\rhd ((\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
*(-u1
)
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{z}}
*(-u1
)
)
)
\rhd (\underline{
\mathsf{x}}
*(\underline{
\mathsf{z}}
*(-u1
)
)
)
 = (\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
\gg (-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = (\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
; EqAdditionLeft
\rhd (-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
 = (\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
\gg ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
)
 = ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
)
; DistributionOut
\gg ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
)
; eqTransitivity
\rhd ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
)
 = ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
)
\rhd ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(\underline{
\mathsf{x}}
*(-u\underline{
\mathsf{z}}
)
)
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
)
\gg ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[DistributionOut(Minus)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon ((\underline{
\mathsf{x}}
*\underline{
\mathsf{y}}
)
+(-u(\underline{
\mathsf{x}}
*\underline{
\mathsf{z}}
)
)
)
 = (\underline{
\mathsf{x}}
*(\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
)
]$

$\rm
[DistributionOut(Minus)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}DistributionOut(Minus)
\mbox{''}
]$

$\rm
[DistributionOut(Minus)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distributionOut(Minus)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:32}{}\section* {$\rm PositiveToRight(Eq)(1 term)$}

$\rm
[PositiveToRight(Eq)(1 term)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\vdash eqSymmetry
\rhd \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\gg \underline{
\mathsf{y}}
 = \underline{
\mathsf{x}}
; PositiveToLeft(Eq)(1 term)
\rhd \underline{
\mathsf{y}}
 = \underline{
\mathsf{x}}
\gg (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
 = 0
; eqSymmetry
\rhd (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
 = 0
\gg 0
 = (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveToRight(Eq)(1 term)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
\vdash 0
 = (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
]$

$\rm
[PositiveToRight(Eq)(1 term)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$

$\rm
[PositiveToRight(Eq)(1 term)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:33}{}\section* {$\rm SameSeries(NumDiff)$}

$\rm
[SameSeries(NumDiff)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{
\mathsf {o}}
\colon 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon \underline{
\mathsf {o}}
 = \underline{
\mathsf {p}}
\vdash \underline{(n1)}
 = \underline{(n2)}
\vdash SameSeries
\rhd \underline{
\mathsf {o}}
 = \underline{
\mathsf {p}}
\gg \underline{(fx)}
[\underline{
\mathsf {o}}
]
 = \underline{(fx)}
[\underline{
\mathsf {p}}
]
; SameSeries
\rhd \underline{(n1)}
 = \underline{(n2)}
\gg \underline{(fy)}
[\underline{(n1)}
]
 = \underline{(fy)}
[\underline{(n2)}
]
; EqNegated
\rhd \underline{(fy)}
[\underline{(n1)}
]
 = \underline{(fy)}
[\underline{(n2)}
]
\gg (-u\underline{(fy)}
[\underline{(n1)}
]
)
 = (-u\underline{(fy)}
[\underline{(n2)}
]
)
; AddEquations
\rhd \underline{(fx)}
[\underline{
\mathsf {o}}
]
 = \underline{(fx)}
[\underline{
\mathsf {p}}
]
\rhd (-u\underline{(fy)}
[\underline{(n1)}
]
)
 = (-u\underline{(fy)}
[\underline{(n2)}
]
)
\gg (\underline{(fx)}
[\underline{
\mathsf {o}}
]
+(-u\underline{(fy)}
[\underline{(n1)}
]
)
)
 = (\underline{(fx)}
[\underline{
\mathsf {p}}
]
+(-u\underline{(fy)}
[\underline{(n2)}
]
)
)
; SameNumerical
\rhd (\underline{(fx)}
[\underline{
\mathsf {o}}
]
+(-u\underline{(fy)}
[\underline{(n1)}
]
)
)
 = (\underline{(fx)}
[\underline{
\mathsf {p}}
]
+(-u\underline{(fy)}
[\underline{(n2)}
]
)
)
\gg |(\underline{(fx)}
[\underline{
\mathsf {o}}
]
+(-u\underline{(fy)}
[\underline{(n1)}
]
)
)
|
 = |(\underline{(fx)}
[\underline{
\mathsf {p}}
]
+(-u\underline{(fy)}
[\underline{(n2)}
]
)
)
|
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameSeries(NumDiff)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{
\mathsf {o}}
\colon 
\forall \underline{
\mathsf {p}}
\colon 
\forall \underline{(n1)}
\colon 
\forall \underline{(n2)}
\colon \underline{
\mathsf {o}}
 = \underline{
\mathsf {p}}
\vdash \underline{(n1)}
 = \underline{(n2)}
\vdash |(\underline{(fx)}
[\underline{
\mathsf {o}}
]
+(-u\underline{(fy)}
[\underline{(n1)}
]
)
)
|
 = |(\underline{(fx)}
[\underline{
\mathsf {p}}
]
+(-u\underline{(fy)}
[\underline{(n2)}
]
)
)
|
]$

$\rm
[SameSeries(NumDiff)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameSeries(NumDiff)
\mbox{''}
]$

$\rm
[SameSeries(NumDiff)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameSeries(NumDiff)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:34}{}\section* {$\rm ToNegatedDoubleImply$}

$\rm
[ToNegatedDoubleImply
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash MP2
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\rhd \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {b}}
\gg \underline{
\mathsf {c}}
; FromContradiction
\rhd \underline{
\mathsf {c}}
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
; \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\vdash MP3
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\rhd \underline{
\mathsf {a}}
\rhd \underline{
\mathsf {b}}
\rhd \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
; 
\mbox{prop lemma imply negation}
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\gg \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[ToNegatedDoubleImply
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\vdash \dot{\neg}\,(\underline{
\mathsf {c}}
)n
\vdash \dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
)n
]$

$\rm
[ToNegatedDoubleImply
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToNegatedDoubleImply
\mbox{''}
]$

$\rm
[ToNegatedDoubleImply
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}imply
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:35}{}\section* {$\rm AddNegatedAll$}

$\rm
[AddNegatedAll
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\vdash AddAll 
\rhd \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\gg 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {b}}
\Rightarrow 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
; Contrapositive
\rhd 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {b}}
\Rightarrow 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {b}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[AddNegatedAll
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {b}}
)n
]$

$\rm
[AddNegatedAll
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AddNegatedAll
\mbox{''}
]$

$\rm
[AddNegatedAll
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}addNegatedAll
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:36}{}\section* {$\rm (A)to(~E~)(Imply)$}

$\rm
[(A)to(~E~)(Imply)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\vdash A4
\mathop {\char64} \underline{(v1)}
\rhd 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
; AddDoubleNeg
\rhd \underline{
\mathsf {a}}
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
; Gen
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\gg 
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
; AddDoubleNeg
\rhd 
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
; 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon Ded
\rhd 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
\gg 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[(A)to(~E~)(Imply)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
]$

$\rm
[(A)to(~E~)(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(A)to(
\char126 E
\char126 )(Imply)
\mbox{''}
]$

$\rm
[(A)to(~E~)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(A)to(
\char126 E
\char126 )(Imply)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:37}{}\section* {$\rm (E)to(~A~)(Imply)$}

$\rm
[(E)to(~A~)(Imply)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon AutoImply
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
; Repetition
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[(E)to(~A~)(Imply)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
]$

$\rm
[(E)to(~A~)(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(E)to(
\char126 A
\char126 )(Imply)
\mbox{''}
]$

$\rm
[(E)to(~A~)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(E)to(
\char126 A
\char126 )(Imply)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:38}{}\section* {$\rm (E~)to(~A)(Imply)$}

$\rm
[(E~)to(~A)(Imply)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\vdash AddDoubleNeg
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
; (A)to(~E~)(Imply)
\gg 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
; MT
\rhd 
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
)n
; 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon Ded
\rhd 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[(E~)to(~A)(Imply)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \underline{
\mathsf {a}}
)n
]$

$\rm
[(E~)to(~A)(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(E
\char126 )to(
\char126 A)(Imply)
\mbox{''}
]$

$\rm
[(E~)to(~A)(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(E
\char126 )to(
\char126 A)(Imply)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:39}{}\section* {$\rm ToNegatedAEA $}

$\rm
[ToNegatedAEA 
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{(v2)}
\colon 
\forall \underline{(v3)}
\colon 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\vdash (E~)to(~A)(Imply)
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
; AddNegatedAll
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
; (E)to(~A~)(Imply)
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
; ImplyTransitivity
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
; AddNegatedAll
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
)n
; (E)to(~A~)(Imply)
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
; ImplyTransitivity
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
)n
; MP
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[ToNegatedAEA 
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{(v2)}
\colon 
\forall \underline{(v3)}
\colon 
\forall \underline{
\mathsf {a}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \underline{
\mathsf {a}}
)n
)n
)n
]$

$\rm
[ToNegatedAEA 
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToNegatedAEA
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{''}
]$

$\rm
[ToNegatedAEA 
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}toNegatedAEA
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:40}{}\section* {$\rm Three2threeTerms(R)$}

$\rm
[Three2threeTerms(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Three2twoTerms(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusAssociativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[Three2threeTerms(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[Three2threeTerms(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Three2threeTerms(R)
\mbox{''}
]$

$\rm
[Three2threeTerms(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2threeTerms(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:41}{}\section* {$\rm LessNeq(F)(Helper)$}

$\rm
[LessNeq(F)(Helper)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\vdash A4
\mathop {\char64} \underline{
\mathsf {n}}
\rhd 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
; FirstConjunct
\rhd \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\gg \dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
; SecondConjunct
\rhd \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\gg \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
; leqReflexivity
\gg \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
; MP
\rhd \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
\rhd \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\gg \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
; SwitchTerms(x<=y-z)
\rhd \underline{(fx)}
[\underline{
\mathsf {n}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(\epsilon)}
)
)
\gg \underline{(\epsilon)}
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
; x<=|x|
\gg (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
 <= |(\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
|
; leqTransitivity
\rhd \underline{(\epsilon)}
 <= (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
\rhd (\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
 <= |(\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
|
\gg \underline{(\epsilon)}
 <= |(\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
|
; NumericalDifference
\gg |(\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
|
 = |(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
; subLeqRight
\rhd |(\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
|
 = |(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
\rhd \underline{(\epsilon)}
 <= |(\underline{(fy)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fx)}
[\underline{
\mathsf {n}}
]
)
)
|
\gg \underline{(\epsilon)}
 <= |(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
; toNotLess
\rhd \underline{(\epsilon)}
 <= |(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
\gg \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
; ToNegatedDoubleImply
\rhd \dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\rhd \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\rhd \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
\gg 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LessNeq(F)(Helper)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
]$

$\rm
[LessNeq(F)(Helper)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessNeq(F)(Helper)
\mbox{''}
]$

$\rm
[LessNeq(F)(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNeq(F)
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:42}{}\section* {$\rm LessNeq(F)$}

$\rm
[LessNeq(F)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash Repetition
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; Ded
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; LessNeq(F)(Helper)
\gg 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
; TwiceExistMP
\rhd 
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\underline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
; IntroExist
\mathop {\char64} \underline{
\mathsf {n}}
\rhd \dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {n}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {n}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {n}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; Gen
\rhd \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg 
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; IntroExist
\mathop {\char64} \underline{(\epsilon)}
\rhd 
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
)n
)n
; ToNegatedAEA 
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; Ded
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \underline{(\epsilon)}
)n
)n
)n
\Rightarrow \underline{
\mathsf {n}}
 <= \underline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 <= \underline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
)
|
 = \underline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; Repetition
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LessNeq(F)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
]$

$\rm
[LessNeq(F)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessNeq(F)
\mbox{''}
]$

$\rm
[LessNeq(F)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNeq(F)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:43}{}\section* {$\rm LessNeq(R)$}

$\rm
[LessNeq(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash Repetition
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; LessNeq(F)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; To!!==
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LessNeq(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
]$

$\rm
[LessNeq(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessNeq(R)
\mbox{''}
]$

$\rm
[LessNeq(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNeq(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:44}{}\section* {$\rm x=x+(y-y)(R)$}

$\rm
[x=x+(y-y)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Plus0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqAdditionLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[x=x+(y-y)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[x=x+(y-y)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{=}x
{+}(y
\mbox{-}y)(R)
\mbox{''}
]$

$\rm
[x=x+(y-y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{=}x
{+}(y
\mbox{-}y)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:45}{}\section* {$\rm x=x+y-y(R)$}

$\rm
[x=x+y-y(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon x=x+(y-y)(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusAssociativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[x=x+y-y(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[x=x+y-y(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{=}x
{+}y
\mbox{-}y(R)
\mbox{''}
]$

$\rm
[x=x+y-y(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{=}x
{+}y
\mbox{-}y(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:46}{}\section* {$\rm SubtractEquations(R)$}

$\rm
[SubtractEquations(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PositiveToRight(Eq)(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqAdditionLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusAssociativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SubtractEquations(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SubtractEquations(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubtractEquations(R)
\mbox{''}
]$

$\rm
[SubtractEquations(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquations(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:47}{}\section* {$\rm NeqAddition(R)$}

$\rm
[NeqAddition(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash ==\!{}Reflexivity
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubtractEquations(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromContradiction
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash MP
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\mbox{prop lemma imply negation}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NeqAddition(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
]$

$\rm
[NeqAddition(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NeqAddition(R)
\mbox{''}
]$

$\rm
[NeqAddition(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}neqAddition(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:48}{}\section* {$\rm PositiveToRight(Less)(R)$}

$\rm
[PositiveToRight(Less)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash lessAddition(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLessLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveToRight(Less)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[PositiveToRight(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToRight(Less)(R)
\mbox{''}
]$

$\rm
[PositiveToRight(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:49}{}\section* {$\rm PositiveToRight(Less)(1 term)(R)$}

$\rm
[PositiveToRight(Less)(1 term)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash Plus0Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLessLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; PositiveToRight(Less)(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveToRight(Less)(1 term)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[PositiveToRight(Less)(1 term)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

$\rm
[PositiveToRight(Less)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:50}{}\section* {$\rm LeqNeqLess(R)$}

$\rm
[LeqNeqLess(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash LeqAntisymmetry(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromContradiction
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash MP2
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\mbox{prop lemma imply negation}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; ToLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LeqNeqLess(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[LeqNeqLess(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqNeqLess(R)
\mbox{''}
]$

$\rm
[LeqNeqLess(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqNeqLess(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:51}{}\section* {$\rm SubLeqLeft(R)$}

$\rm
[SubLeqLeft(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\mbox{lemma eqLeq(R)}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqTransitivity(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SubLeqLeft(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SubLeqLeft(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubLeqLeft(R)
\mbox{''}
]$

$\rm
[SubLeqLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subLeqLeft(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:52}{}\section* {$\rm ToLeq(Advanced)(R)$}

$\rm
[ToLeq(Advanced)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fep)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash PositiveToRight(Less)(1 term)(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Three2threeTerms(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; JoinConjuncts
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
; IntroExist
\mathop {\char64}  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
)n
; 
\forall \underline{(fep)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fep)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
)n
; \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash ToNegatedAnd
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
; Gen
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
\gg 
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
; (A~)to(~E)
\rhd 
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
)n
)n
; MT
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
)n
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(fep)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
; FromNotLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[ToLeq(Advanced)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fep)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fep)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fep)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[ToLeq(Advanced)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToLeq(Advanced)(R)
\mbox{''}
]$

$\rm
[ToLeq(Advanced)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}toLeq(Advanced)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:53}{}\section* {$\rm LeqLessTransitivity(R)$}

$\rm
[LeqLessTransitivity(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LessNeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; SubLeqLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqAntisymmetry(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromContradiction
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash MP2
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\mbox{prop lemma imply negation}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqTransitivity(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqNeqLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LeqLessTransitivity(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[LeqLessTransitivity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqLessTransitivity(R)
\mbox{''}
]$

$\rm
[LeqLessTransitivity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqLessTransitivity(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:54}{}\section* {$\rm NegativeToLeft(Eq)(R)$}

$\rm
[NegativeToLeft(Eq)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Three2threeTerms(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeToLeft(Eq)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[NegativeToLeft(Eq)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToLeft(Eq)(R)
\mbox{''}
]$

$\rm
[NegativeToLeft(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Eq)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:55}{}\section* {$\rm NegativeToRight(Less)(R)$}

$\rm
[NegativeToRight(Less)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash lessAddition(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; Three2threeTerms(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLessLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeToRight(Less)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[NegativeToRight(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToRight(Less)(R)
\mbox{''}
]$

$\rm
[NegativeToRight(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToRight(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:56}{}\section* {$\rm !!==Symmetry$}

$\rm
[!!==Symmetry
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash MT
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[!!==Symmetry
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
]$

$\rm
[!!==Symmetry
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}!!
{=}
{=}Symmetry
\mbox{''}
]$

$\rm
[!!==Symmetry
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}!!
{=}
{=}Symmetry
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:57}{}\section* {$\rm SwitchTerms(x<=y-z)$}

$\rm
[SwitchTerms(x<=y-z)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
\vdash negativeToLeft(Leq)
\rhd \underline{
\mathsf{x}}
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
\gg (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 <= \underline{
\mathsf{y}}
; plusCommutativity
\gg (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{z}}
+\underline{
\mathsf{x}}
)
; subLeqLeft
\rhd (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 = (\underline{
\mathsf{z}}
+\underline{
\mathsf{x}}
)
\rhd (\underline{
\mathsf{x}}
+\underline{
\mathsf{z}}
)
 <= \underline{
\mathsf{y}}
\gg (\underline{
\mathsf{z}}
+\underline{
\mathsf{x}}
)
 <= \underline{
\mathsf{y}}
; PositiveToRight(Leq)
\rhd (\underline{
\mathsf{z}}
+\underline{
\mathsf{x}}
)
 <= \underline{
\mathsf{y}}
\gg \underline{
\mathsf{z}}
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SwitchTerms(x<=y-z)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon 
\forall \underline{
\mathsf{z}}
\colon \underline{
\mathsf{x}}
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{z}}
)
)
\vdash \underline{
\mathsf{z}}
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
]$

$\rm
[SwitchTerms(x<=y-z)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SwitchTerms(x
{<}
{=}y
\mbox{-}z)
\mbox{''}
]$

$\rm
[SwitchTerms(x<=y-z)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}switchTerms(x
{<}
{=}y
\mbox{-}z)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:58}{}\section* {$\rm Plus0Left(R)$}

$\rm
[Plus0Left(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon Plus0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[Plus0Left(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[Plus0Left(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Plus0Left(R)
\mbox{''}
]$

$\rm
[Plus0Left(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}plus0Left(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:59}{}\section* {$\rm PositiveToRight(Eq)(R)$}

$\rm
[PositiveToRight(Eq)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveToRight(Eq)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[PositiveToRight(Eq)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToRight(Eq)(R)
\mbox{''}
]$

$\rm
[PositiveToRight(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Eq)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:60}{}\section* {$\rm EqAdditionLeft(R)$}

$\rm
[EqAdditionLeft(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[EqAdditionLeft(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[EqAdditionLeft(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqAdditionLeft(R)
\mbox{''}
]$

$\rm
[EqAdditionLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eqAdditionLeft(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:61}{}\section* {$\rm Three2twoTerms(R)$}

$\rm
[Three2twoTerms(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAdditionLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusAssociativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[Three2twoTerms(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[Three2twoTerms(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Three2twoTerms(R)
\mbox{''}
]$

$\rm
[Three2twoTerms(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2twoTerms(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:62}{}\section* {$\rm To!!==$}

$\rm
[To!!==
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash From==
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\vdash MT
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[To!!==
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {m}}
\colon 
\forall \underline{
\mathsf {n}}
\colon 
\forall \underline{(\epsilon)}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
]$

$\rm
[To!!==
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}To!!
{=}
{=}
\mbox{''}
]$

$\rm
[To!!==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to!!
{=}
{=}
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:63}{}\section* {$\rm PositiveToRight(Less)(1 term)$}

$\rm
[PositiveToRight(Less)(1 term)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
)n
)n
)n
\vdash plus0Left
\gg (0
+\underline{
\mathsf{x}}
)
 = \underline{
\mathsf{x}}
; eqSymmetry
\rhd (0
+\underline{
\mathsf{x}}
)
 = \underline{
\mathsf{x}}
\gg \underline{
\mathsf{x}}
 = (0
+\underline{
\mathsf{x}}
)
; SubLessLeft
\rhd \underline{
\mathsf{x}}
 = (0
+\underline{
\mathsf{x}}
)
\rhd \dot{\neg}\,(\underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
)n
)n
)n
\gg \dot{\neg}\,((0
+\underline{
\mathsf{x}}
)
 <= \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,((0
+\underline{
\mathsf{x}}
)
 = \underline{
\mathsf{y}}
)n
)n
)n
; PositiveToRight(Less)
\rhd \dot{\neg}\,((0
+\underline{
\mathsf{x}}
)
 <= \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,((0
+\underline{
\mathsf{x}}
)
 = \underline{
\mathsf{y}}
)n
)n
)n
\gg \dot{\neg}\,(0
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveToRight(Less)(1 term)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf{x}}
\colon 
\forall \underline{
\mathsf{y}}
\colon \dot{\neg}\,(\underline{
\mathsf{x}}
 <= \underline{
\mathsf{y}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf{x}}
 = \underline{
\mathsf{y}}
)n
)n
)n
\vdash \dot{\neg}\,(0
 <= (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = (\underline{
\mathsf{y}}
+(-u\underline{
\mathsf{x}}
)
)
)n
)n
)n
]$

$\rm
[PositiveToRight(Less)(1 term)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$

$\rm
[PositiveToRight(Less)(1 term)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:64}{}\section* {$\rm (A~)to(~E)$}

$\rm
[(A~)to(~E)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash AddDoubleNeg
\rhd 
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[(A~)to(~E)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
)n
]$

$\rm
[(A~)to(~E)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(A
\char126 )to(
\char126 E)
\mbox{''}
]$

$\rm
[(A~)to(~E)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(A
\char126 )to(
\char126 E)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:65}{}\section* {$\rm NegativeToRight(Eq)(R)$}

$\rm
[NegativeToRight(Eq)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Three2threeTerms(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeToRight(Eq)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[NegativeToRight(Eq)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToRight(Eq)(R)
\mbox{''}
]$

$\rm
[NegativeToRight(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToRight(Eq)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:66}{}\section* {$\rm NegativeToRight(Eq)(1 term)(R)$}

$\rm
[NegativeToRight(Eq)(1 term)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash NegativeToRight(Eq)(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeToRight(Eq)(1 term)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[NegativeToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

$\rm
[NegativeToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:67}{}\section* {$\rm DoubleMinus(R)$}

$\rm
[DoubleMinus(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; NegativeToRight(Eq)(1 term)(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[DoubleMinus(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[DoubleMinus(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}DoubleMinus(R)
\mbox{''}
]$

$\rm
[DoubleMinus(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}doubleMinus(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:68}{}\section* {$\rm UniqueNegative(R)$}

$\rm
[UniqueNegative(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PositiveToRight(Eq)(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PositiveToRight(Eq)(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[UniqueNegative(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[UniqueNegative(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}UniqueNegative(R)
\mbox{''}
]$

$\rm
[UniqueNegative(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}uniqueNegative(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:69}{}\section* {$\rm SubtractEquationsLeft(R)$}

$\rm
[SubtractEquationsLeft(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubtractEquations(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fz)}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fu)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SubtractEquationsLeft(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SubtractEquationsLeft(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubtractEquationsLeft(R)
\mbox{''}
]$

$\rm
[SubtractEquationsLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquationsLeft(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:70}{}\section* {$\rm EqNegated(R)$}

$\rm
[EqNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubtractEquationsLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[EqNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[EqNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}EqNegated(R)
\mbox{''}
]$

$\rm
[EqNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}eqNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:71}{}\section* {$\rm NeqNegated(R)$}

$\rm
[NeqNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqNegated(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; DoubleMinus(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; DoubleMinus(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromContradiction
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash MP
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\mbox{prop lemma imply negation}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NeqNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
]$

$\rm
[NeqNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NeqNegated(R)
\mbox{''}
]$

$\rm
[NeqNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}neqNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:72}{}\section* {$\rm -0=0(R)$}

$\rm
[-0=0(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; UniqueNegative(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[-0=0(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[-0=0(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox{-}0
{=}0(R)
\mbox{''}
]$

$\rm
[-0=0(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}0
{=}0(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:73}{}\section* {$\rm NegativeNegated(R)$}

$\rm
[NegativeNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash LessNegated(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; -0=0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLessLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[NegativeNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeNegated(R)
\mbox{''}
]$

$\rm
[NegativeNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:74}{}\section* {$\rm FromLeqGeq(R)$}

$\rm
[FromLeqGeq(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \underline{
\mathsf {a}}
\vdash LeqTotality(R)
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromDisjuncts
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \underline{
\mathsf {a}}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \underline{
\mathsf {a}}
\gg \underline{
\mathsf {a}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[FromLeqGeq(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {a}}
]$

$\rm
[FromLeqGeq(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromLeqGeq(R)
\mbox{''}
]$

$\rm
[FromLeqGeq(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}leqGeq(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:75}{}\section* {$\rm 0<=|x|(R)$}

$\rm
[0<=|x|(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash NonnegativeNumerical(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqRight(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash ToLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; NegativeNumerical(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; NegativeNegated(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqRight(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromNegations
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[0<=|x|(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[0<=|x|(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0
{<}
{=}|x|(R)
\mbox{''}
]$

$\rm
[0<=|x|(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}
{=}|x|(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:76}{}\section* {$\rm PositiveNegated(R)$}

$\rm
[PositiveNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash LessNegated(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; -0=0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLessRight(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[PositiveNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveNegated(R)
\mbox{''}
]$

$\rm
[PositiveNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:77}{}\section* {$\rm AddEquations(R)$}

$\rm
[AddEquations(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqAdditionLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[AddEquations(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon 
\forall \underline{(fu)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fu)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fu)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[AddEquations(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}AddEquations(R)
\mbox{''}
]$

$\rm
[AddEquations(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}addEquations(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:78}{}\section* {$\rm Times(-1)(R)$}

$\rm
[Times(-1)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqMultiplicationLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x*0=0(R)(fff)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Distribution(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PositiveToRight(Eq)(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Times1(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqNegated(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[Times(-1)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[Times(-1)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Times(
\mbox{-}1)(R)
\mbox{''}
]$

$\rm
[Times(-1)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times(
\mbox{-}1)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:79}{}\section* {$\rm Times(-1)Left(R)$}

$\rm
[Times(-1)Left(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon Times(-1)(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; TimesCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[Times(-1)Left(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[Times(-1)Left(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Times(
\mbox{-}1)Left(R)
\mbox{''}
]$

$\rm
[Times(-1)Left(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times(
\mbox{-}1)Left(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:80}{}\section* {$\rm -x-y=-(x+y)(R)$}

$\rm
[-x-y=-(x+y)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Times(-1)Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Times(-1)Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; AddEquations(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; DistributionOut(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Times(-1)Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[-x-y=-(x+y)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[-x-y=-(x+y)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox{-}x
\mbox{-}y
{=}
\mbox{-}(x
{+}y)(R)
\mbox{''}
]$

$\rm
[-x-y=-(x+y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}x
\mbox{-}y
{=}
\mbox{-}(x
{+}y)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:81}{}\section* {$\rm LessTotality(R)$}

$\rm
[LessTotality(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash FromNotLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; !!==Symmetry
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; LeqNeqLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LessTotality(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[LessTotality(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessTotality(R)
\mbox{''}
]$

$\rm
[LessTotality(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessTotality(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:82}{}\section* {$\rm SameNumerical(R)$}

$\rm
[SameNumerical(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash NonnegativeNumerical(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqRight(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; NonnegativeNumerical(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash ToLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; NegativeNumerical(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLessLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; NegativeNumerical(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqNegated(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromNegations
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; MP
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SameNumerical(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fy)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SameNumerical(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameNumerical(R)
\mbox{''}
]$

$\rm
[SameNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameNumerical(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:83}{}\section* {$\rm MinusNegated(R)$}

$\rm
[MinusNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon DoubleMinus(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; -x-y=-(x+y)(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqNegated(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[MinusNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[MinusNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MinusNegated(R)
\mbox{''}
]$

$\rm
[MinusNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}minusNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:84}{}\section* {$\rm PositiveNumerical(R)$}

$\rm
[PositiveNumerical(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; NonnegativeNumerical(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[PositiveNumerical(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[PositiveNumerical(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveNumerical(R)
\mbox{''}
]$

$\rm
[PositiveNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveNumerical(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:85}{}\section* {$\rm SignNumerical(+)(R)$}

$\rm
[SignNumerical(+)(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash PositiveNumerical(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PositiveNegated(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; NegativeNumerical(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; DoubleMinus(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SignNumerical(+)(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SignNumerical(+)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SignNumerical(
{+})(R)
\mbox{''}
]$

$\rm
[SignNumerical(+)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}signNumerical(
{+})(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:86}{}\section* {$\rm NonnegativeNumerical(R)$}

$\rm
[NonnegativeNumerical(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash from<<==
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; NonnegativeNumerical(F)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg |f\underline{(fx)}
|
 = \underline{(fx)}
; (Adgic)SameR
\rhd |f\underline{(fx)}
|
 = \underline{(fx)}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Repetition
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NonnegativeNumerical(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[NonnegativeNumerical(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonnegativeNumerical(R)
\mbox{''}
]$

$\rm
[NonnegativeNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonnegativeNumerical(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:87}{}\section* {$\rm NegativeNumerical(R)$}

$\rm
[NegativeNumerical(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash FromLess(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; to<<==
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromContradiction
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; 
\forall \underline{(fx)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash MP
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; 
\mbox{prop lemma imply negation}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
; NegativeNumerical(F)
\gg |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
; (Adgic)SameR
\rhd |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Repetition
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeNumerical(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[NegativeNumerical(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeNumerical(R)
\mbox{''}
]$

$\rm
[NegativeNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeNumerical(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:88}{}\section* {$\rm LeqNegated(R)$}

$\rm
[LeqNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash leqAddition(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusCommutativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqRight(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; leqAddition(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x=x+y-y(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SubLeqRight(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LeqNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[LeqNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqNegated(R)
\mbox{''}
]$

$\rm
[LeqNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:89}{}\section* {$\rm LessNegated(R)$}

$\rm
[LessNegated(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqNegated(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LessNeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; NeqNegated(R)
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; !!==Symmetry
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; LeqNeqLess(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[LessNegated(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
]$

$\rm
[LessNegated(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessNegated(R)
\mbox{''}
]$

$\rm
[LessNegated(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNegated(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:90}{}\section* {$\rm SubLeqRight(R)$}

$\rm
[SubLeqRight(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash 
\mbox{lemma eqLeq(R)}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqTransitivity(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SubLeqRight(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fz)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fz)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SubLeqRight(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubLeqRight(R)
\mbox{''}
]$

$\rm
[SubLeqRight(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subLeqRight(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:91}{}\section* {$\rm FromLess(R)$}

$\rm
[FromLess(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash LessLeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqAntisymmetry(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LessNeq(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; FromContradiction
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash MP
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
; 
\mbox{prop lemma imply negation}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[FromLess(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fy)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
]$

$\rm
[FromLess(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromLess(R)
\mbox{''}
]$

$\rm
[FromLess(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}fromLess(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:92}{}\section* {$\rm DistributionOut(R)$}

$\rm
[DistributionOut(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon Distribution(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[DistributionOut(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall \underline{(fz)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
*\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+\underline{(fz)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[DistributionOut(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}DistributionOut(R)
\mbox{''}
]$

$\rm
[DistributionOut(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distributionOut(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:93}{}\section* {$\rm x*0+x=x(R)$}

$\rm
[x*0+x=x(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon Times1(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqAdditionLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Distribution(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Plus0Left(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqMultiplicationLeft(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity5
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,1
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[x*0+x=x(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[x*0+x=x(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{*}0
{+}x
{=}x(R)
\mbox{''}
]$

$\rm
[x*0+x=x(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{*}0
{+}x
{=}x(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:94}{}\section* {$\rm x*0=0(R)(fff)$}

$\rm
[x*0=0(R)(fff)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon x=x+(y-y)(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; PlusAssociativity(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; x*0+x=x(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; EqAddition(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Negative(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity5
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[x*0=0(R)(fff)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {e}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
* \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[x*0=0(R)(fff)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{*}0
{=}0(R)(fff)
\mbox{''}
]$

$\rm
[x*0=0(R)(fff)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{*}0
{=}0(R)fff
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:95}{}\section* {$\rm SignNumerical(R)$}

$\rm
[SignNumerical(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash NegativeNegated(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; SignNumerical(+)(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; DoubleMinus(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SameNumerical(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash EqNegated(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; -0=0(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity4
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SameNumerical(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash SignNumerical(+)(R)
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LessTotality(R)
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; From3Disjuncts
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[SignNumerical(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[SignNumerical(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SignNumerical(R)
\mbox{''}
]$

$\rm
[SignNumerical(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}signNumerical(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:96}{}\section* {$\rm NumericalDifference(R)$}

$\rm
[NumericalDifference(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon SignNumerical(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; MinusNegated(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; SameNumerical(R)
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; eqTransitivity
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NumericalDifference(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fx)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fy)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {d}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(\underline{(fy)}
[\underline{
\mathsf {m}}
]
+ \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[NumericalDifference(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NumericalDifference(R)
\mbox{''}
]$

$\rm
[NumericalDifference(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}numericalDifference(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:97}{}\section* {$\rm x<=|x|(R)$}

$\rm
[x<=|x|(R)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash NonnegativeNumerical(R)
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; ==\!{}Symmetry
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\mbox{lemma eqLeq(R)}
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash 0<=|x|(R)
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; LeqTransitivity(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromLeqGeq(R)
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= ( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[x<=|x|(R)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(|f\underline{(fx)}
|
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[x<=|x|(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{<}
{=}|x|(R)
\mbox{''}
]$

$\rm
[x<=|x|(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{<}
{=}|x|(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:98}{}\section* {$\rm USlimitIsUpperBound(Helper)$}

$\rm
[USlimitIsUpperBound(Helper)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USlimitIsUpperBound(Helper)
\mbox{''}
]$

$\rm
[USlimitIsUpperBound(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsUpperBound
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:99}{}\section* {$\rm USlimitIsUpperBound$}

$\rm
[USlimitIsUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USlimitIsUpperBound
\mbox{''}
]$

$\rm
[USlimitIsUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsUpperBound
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:100}{}\section* {$\rm (-1)*(-1)+(-1)*1=0(R)$}

$\rm
[(-1)*(-1)+(-1)*1=0(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox{-}1)
{*}(
\mbox{-}1)
{+}(
\mbox{-}1)
{*}1
{=}0(R)
\mbox{''}
]$

$\rm
[(-1)*(-1)+(-1)*1=0(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(
\mbox{-}1)
{*}(
\mbox{-}1)
{+}(
\mbox{-}1)
{*}1
{=}0(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:101}{}\section* {$\rm (-1)*(-1)=1(R)$}

$\rm
[(-1)*(-1)=1(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(
\mbox{-}1)
{*}(
\mbox{-}1)
{=}1(R)
\mbox{''}
]$

$\rm
[(-1)*(-1)=1(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(
\mbox{-}1)
{*}(
\mbox{-}1)
{=}1(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:102}{}\section* {$\rm 0<1Helper(R)$}

$\rm
[0<1Helper(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0
{<}1Helper(R)
\mbox{''}
]$

$\rm
[0<1Helper(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}1Helper(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:103}{}\section* {$\rm 0<1(R)$}

$\rm
[0<1(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0
{<}1(R)
\mbox{''}
]$

$\rm
[0<1(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}1(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:104}{}\section* {$\rm ExpZero(Exact)(R)$}

$\rm
[ExpZero(Exact)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpZero(Exact)(R)
\mbox{''}
]$

$\rm
[ExpZero(Exact)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expZero
\linebreak [0]\ \hskip0em plus2.0em{}exact(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:105}{}\section* {$\rm PositiveBase(R)(Base)$}

$\rm
[PositiveBase(R)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveBase(R)(Base)
\mbox{''}
]$

$\rm
[PositiveBase(R)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase(R)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:106}{}\section* {$\rm Three2twoFactors(R)$}

$\rm
[Three2twoFactors(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Three2twoFactors(R)
\mbox{''}
]$

$\rm
[Three2twoFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2twoFactors(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:107}{}\section* {$\rm x=x*y*(1/y)(R)$}

$\rm
[x=x*y*(1/y)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{=}x
{*}y
{*}(1/y)(R)
\mbox{''}
]$

$\rm
[x=x*y*(1/y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{=}x
{*}y
{*}(1/y)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:108}{}\section* {$\rm NeqMultiplication(R)$}

$\rm
[NeqMultiplication(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NeqMultiplication(R)
\mbox{''}
]$

$\rm
[NeqMultiplication(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}neqMultiplication(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:109}{}\section* {$\rm LessTransitivity(R)$}

$\rm
[LessTransitivity(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessTransitivity(R)
\mbox{''}
]$

$\rm
[LessTransitivity(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessTransitivity(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:110}{}\section* {$\rm 0<2(R)$}

$\rm
[0<2(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0
{<}2(R)
\mbox{''}
]$

$\rm
[0<2(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}2(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:111}{}\section* {$\rm SameExp(R)(Base)$}

$\rm
[SameExp(R)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameExp(R)(Base)
\mbox{''}
]$

$\rm
[SameExp(R)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp(R)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:112}{}\section* {$\rm SameExp(R)(Indu)$}

$\rm
[SameExp(R)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameExp(R)(Indu)
\mbox{''}
]$

$\rm
[SameExp(R)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp(R)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:113}{}\section* {$\rm SameExp(R)$}

$\rm
[SameExp(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SameExp(R)
\mbox{''}
]$

$\rm
[SameExp(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}sameExp(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:114}{}\section* {$\rm SubNeqLeft(R)$}

$\rm
[SubNeqLeft(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubNeqLeft(R)
\mbox{''}
]$

$\rm
[SubNeqLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subNeqLeft(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:115}{}\section* {$\rm SubNeqRight(R)$}

$\rm
[SubNeqRight(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubNeqRight(R)
\mbox{''}
]$

$\rm
[SubNeqRight(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subNeqRight(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:116}{}\section* {$\rm NonzeroFactors(R)$}

$\rm
[NonzeroFactors(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonzeroFactors(R)
\mbox{''}
]$

$\rm
[NonzeroFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonzeroFactors(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:117}{}\section* {$\rm NonnegativeFactors(R)$}

$\rm
[NonnegativeFactors(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonnegativeFactors(R)
\mbox{''}
]$

$\rm
[NonnegativeFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonnegativeFactors(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:118}{}\section* {$\rm PositiveFactors(R)$}

$\rm
[PositiveFactors(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveFactors(R)
\mbox{''}
]$

$\rm
[PositiveFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveFactors(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:119}{}\section* {$\rm LessDivision(R)$}

$\rm
[LessDivision(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessDivision(R)
\mbox{''}
]$

$\rm
[LessDivision(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessDivision(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:120}{}\section* {$\rm 0<1/2(R)$}

$\rm
[0<1/2(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}0
{<}1/2(R)
\mbox{''}
]$

$\rm
[0<1/2(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}0
{<}1/2(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:121}{}\section* {$\rm PositiveToRight(Eq)(1 term)(R)$}

$\rm
[PositiveToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

$\rm
[PositiveToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:122}{}\section* {$\rm Exp(+1)(R)$}

$\rm
[Exp(+1)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Exp(
{+}1)(R)
\mbox{''}
]$

$\rm
[Exp(+1)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}exp(
{+}1)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:123}{}\section* {$\rm PositiveBase(R)(Indu)$}

$\rm
[PositiveBase(R)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveBase(R)(Indu)
\mbox{''}
]$

$\rm
[PositiveBase(R)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase(R)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:124}{}\section* {$\rm PositiveBase(R)$}

$\rm
[PositiveBase(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveBase(R)
\mbox{''}
]$

$\rm
[PositiveBase(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveBase(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:125}{}\section* {$\rm -x*y=-(x*y)(R)$}

$\rm
[-x*y=-(x*y)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}
\mbox{-}x
{*}y
{=}
\mbox{-}(x
{*}y)(R)
\mbox{''}
]$

$\rm
[-x*y=-(x*y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}
\mbox{-}x
{*}y
{=}
\mbox{-}(x
{*}y)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:126}{}\section* {$\rm PositiveToLeft(Eq)(R)$}

$\rm
[PositiveToLeft(Eq)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveToLeft(Eq)(R)
\mbox{''}
]$

$\rm
[PositiveToLeft(Eq)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveToLeft(Eq)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:127}{}\section* {$\rm Times1Left(R)$}

$\rm
[Times1Left(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Times1Left(R)
\mbox{''}
]$

$\rm
[Times1Left(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}times1Left(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:128}{}\section* {$\rm x+x=2*x(R)$}

$\rm
[x+x=2*x(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{+}x
{=}2
{*}x(R)
\mbox{''}
]$

$\rm
[x+x=2*x(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{+}x
{=}2
{*}x(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:129}{}\section* {$\rm (1/2)x+(1/2)x=x(R)$}

$\rm
[(1/2)x+(1/2)x=x(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(1/2)x
{+}(1/2)x
{=}x(R)
\mbox{''}
]$

$\rm
[(1/2)x+(1/2)x=x(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(1/2)x
{+}(1/2)x
{=}x(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:130}{}\section* {$\rm DistributionOut(Minus)(R)$}

$\rm
[DistributionOut(Minus)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}DistributionOut(Minus)(R)
\mbox{''}
]$

$\rm
[DistributionOut(Minus)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}distributionOut(Minus)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:131}{}\section* {$\rm (1/2)(x+y)-x=(1/2)(y-x)(R)$}

$\rm
[(1/2)(x+y)-x=(1/2)(y-x)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]$

$\rm
[(1/2)(x+y)-x=(1/2)(y-x)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}(1/2)(x
{+}y)
\mbox{-}x
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:132}{}\section* {$\rm IntervalSize(R)(Base)$}

$\rm
[IntervalSize(R)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IntervalSize(R)(Base)
\mbox{''}
]$

$\rm
[IntervalSize(R)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(R)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:133}{}\section* {$\rm LessMultiplicationLeft(R)$}

$\rm
[LessMultiplicationLeft(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessMultiplicationLeft(R)
\mbox{''}
]$

$\rm
[LessMultiplicationLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessMultiplicationLeft(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:134}{}\section* {$\rm NegativeToLeft(Less)(R)$}

$\rm
[NegativeToLeft(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToLeft(Less)(R)
\mbox{''}
]$

$\rm
[NegativeToLeft(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:135}{}\section* {$\rm NegativeToLeft(Less)(1 term)(R)$}

$\rm
[NegativeToLeft(Less)(1 term)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

$\rm
[NegativeToLeft(Less)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeToLeft(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:136}{}\section* {$\rm y-(1/2)(x+y)=(1/2)(y-x)(R)$}

$\rm
[y-(1/2)(x+y)=(1/2)(y-x)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]$

$\rm
[y-(1/2)(x+y)=(1/2)(y-x)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}y
\mbox{-}(1/2)(x
{+}y)
{=}(1/2)(y
\mbox{-}x)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:137}{}\section* {$\rm IntervalSize(R)(Indu)$}

$\rm
[IntervalSize(R)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IntervalSize(R)(Indu)
\mbox{''}
]$

$\rm
[IntervalSize(R)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(R)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:138}{}\section* {$\rm IntervalSize(R)$}

$\rm
[IntervalSize(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IntervalSize(R)
\mbox{''}
]$

$\rm
[IntervalSize(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:139}{}\section* {$\rm XSlessUS(R)$}

$\rm
[XSlessUS(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}XSlessUS(R)
\mbox{''}
]$

$\rm
[XSlessUS(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}XSlessUS(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:140}{}\section* {$\rm USdecreasing(+1)(R)$}

$\rm
[USdecreasing(+1)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USdecreasing(
{+}1)(R)
\mbox{''}
]$

$\rm
[USdecreasing(+1)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}1)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:141}{}\section* {$\rm ExpUnbounded(Base)$}

$\rm
[ExpUnbounded(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpUnbounded(Base)
\mbox{''}
]$

$\rm
[ExpUnbounded(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:142}{}\section* {$\rm ExpUnbounded(Indu)$}

$\rm
[ExpUnbounded(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpUnbounded(Indu)
\mbox{''}
]$

$\rm
[ExpUnbounded(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:143}{}\section* {$\rm ExpUnbounded$}

$\rm
[ExpUnbounded
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpUnbounded
\mbox{''}
]$

$\rm
[ExpUnbounded
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expUnbounded
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:144}{}\section* {$\rm 1<=x+1(N)$}

$\rm
[1<=x+1(N)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}1
{<}
{=}x
{+}1(N)
\mbox{''}
]$

$\rm
[1<=x+1(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}1
{<}
{=}x
{+}1(N)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:145}{}\section* {$\rm ExpNonzero(Base)$}

$\rm
[ExpNonzero(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpNonzero(Base)
\mbox{''}
]$

$\rm
[ExpNonzero(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:146}{}\section* {$\rm ExpNonzero(Indu)$}

$\rm
[ExpNonzero(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpNonzero(Indu)
\mbox{''}
]$

$\rm
[ExpNonzero(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:147}{}\section* {$\rm ExpNonzero$}

$\rm
[ExpNonzero
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpNonzero
\mbox{''}
]$

$\rm
[ExpNonzero
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:148}{}\section* {$\rm ExpNonzero(2)$}

$\rm
[ExpNonzero(2)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExpNonzero(2)
\mbox{''}
]$

$\rm
[ExpNonzero(2)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}expNonzero(2)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:149}{}\section* {$\rm HalfBase(Base)$}

$\rm
[HalfBase(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HalfBase(Base)
\mbox{''}
]$

$\rm
[HalfBase(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}halfBase
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:150}{}\section* {$\rm HalfBase(Indu)$}

$\rm
[HalfBase(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HalfBase(Indu)
\mbox{''}
]$

$\rm
[HalfBase(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}halfBase
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:151}{}\section* {$\rm MultiplyEquations(R)$}

$\rm
[MultiplyEquations(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}MultiplyEquations(R)
\mbox{''}
]$

$\rm
[MultiplyEquations(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}multiplyEquations(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:152}{}\section* {$\rm NonreciprocalToRight(Eq)(1 term)(R)$}

$\rm
[NonreciprocalToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonreciprocalToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

$\rm
[NonreciprocalToRight(Eq)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonreciprocalToRight(Eq)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:153}{}\section* {$\rm PositiveNonzero(R)$}

$\rm
[PositiveNonzero(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveNonzero(R)
\mbox{''}
]$

$\rm
[PositiveNonzero(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveNonzero(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:154}{}\section* {$\rm NonzeroProduct(2)(R)$}

$\rm
[NonzeroProduct(2)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonzeroProduct(2)(R)
\mbox{''}
]$

$\rm
[NonzeroProduct(2)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonzeroProduct(2)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:155}{}\section* {$\rm HalfBase$}

$\rm
[HalfBase
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}HalfBase
\mbox{''}
]$

$\rm
[HalfBase
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}halfBase
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:156}{}\section* {$\rm Three2threeFactors(R)$}

$\rm
[Three2threeFactors(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}Three2threeFactors(R)
\mbox{''}
]$

$\rm
[Three2threeFactors(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}three2threeFactors(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:157}{}\section* {$\rm x*y=zBackwards(R)$}

$\rm
[x*y=zBackwards(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}x
{*}y
{=}zBackwards(R)
\mbox{''}
]$

$\rm
[x*y=zBackwards(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}x
{*}y
{=}zBackwards(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:158}{}\section* {$\rm PositiveInverted(R)$}

$\rm
[PositiveInverted(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}PositiveInverted(R)
\mbox{''}
]$

$\rm
[PositiveInverted(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}positiveInverted(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:159}{}\section* {$\rm ReciprocalToRight(Less)(R)$}

$\rm
[ReciprocalToRight(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ReciprocalToRight(Less)(R)
\mbox{''}
]$

$\rm
[ReciprocalToRight(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reciprocalToRight(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:160}{}\section* {$\rm ReciprocalToRight(Less)(1 term)(R)$}

$\rm
[ReciprocalToRight(Less)(1 term)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ReciprocalToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

$\rm
[ReciprocalToRight(Less)(1 term)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}reciprocalToRight(Less)(1
\linebreak [0]\ \hskip0em plus2.0em{}term)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:161}{}\section* {$\rm NonreciprocalToLeft(Less)(R)$}

$\rm
[NonreciprocalToLeft(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonreciprocalToLeft(Less)(R)
\mbox{''}
]$

$\rm
[NonreciprocalToLeft(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonreciprocalToLeft(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:162}{}\section* {$\rm 1<x*y(R)$}

$\rm
[1<x*y(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}1
{<}x
{*}y(R)
\mbox{''}
]$

$\rm
[1<x*y(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}1
{<}x
{*}y(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:163}{}\section* {$\rm SwitchFactors(1/x<y)(R)$}

$\rm
[SwitchFactors(1/x<y)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SwitchFactors(1/x
{<}y)(R)
\mbox{''}
]$

$\rm
[SwitchFactors(1/x<y)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}switchFactors(1/x
{<}y)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:164}{}\section* {$\rm SmallHalving$}

$\rm
[SmallHalving
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SmallHalving
\mbox{''}
]$

$\rm
[SmallHalving
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}smallHalving
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:165}{}\section* {$\rm IntervalSize(anyPositive)$}

$\rm
[IntervalSize(anyPositive)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}IntervalSize(anyPositive)
\mbox{''}
]$

$\rm
[IntervalSize(anyPositive)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}intervalSize(anyPositive)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:166}{}\section* {$\rm USdecreasing(+n)(Base)$}

$\rm
[USdecreasing(+n)(Base)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USdecreasing(
{+}n)(Base)
\mbox{''}
]$

$\rm
[USdecreasing(+n)(Base)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}n)
\linebreak [0]\ \hskip0em plus2.0em{}base
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:167}{}\section* {$\rm USdecreasing(+n)(Indu)$}

$\rm
[USdecreasing(+n)(Indu)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USdecreasing(
{+}n)(Indu)
\mbox{''}
]$

$\rm
[USdecreasing(+n)(Indu)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}n)
\linebreak [0]\ \hskip0em plus2.0em{}indu
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:168}{}\section* {$\rm USdecreasing(+n)$}

$\rm
[USdecreasing(+n)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USdecreasing(
{+}n)
\mbox{''}
]$

$\rm
[USdecreasing(+n)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing(
{+}n)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:169}{}\section* {$\rm USdecreasing$}

$\rm
[USdecreasing
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USdecreasing
\mbox{''}
]$

$\rm
[USdecreasing
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USdecreasing
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:170}{}\section* {$\rm LeqAdditionLeft(R)$}

$\rm
[LeqAdditionLeft(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqAdditionLeft(R)
\mbox{''}
]$

$\rm
[LeqAdditionLeft(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqAdditionLeft(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:171}{}\section* {$\rm ToNotLess(R)$}

$\rm
[ToNotLess(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ToNotLess(R)
\mbox{''}
]$

$\rm
[ToNotLess(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}toNotLess(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:172}{}\section* {$\rm LimitOfUSIsLeq$}

$\rm
[LimitOfUSIsLeq
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LimitOfUSIsLeq
\mbox{''}
]$

$\rm
[LimitOfUSIsLeq
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}limitOfUSIsLeq
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:173}{}\section* {$\rm SubtractEquations(Less)(R)$}

$\rm
[SubtractEquations(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubtractEquations(Less)(R)
\mbox{''}
]$

$\rm
[SubtractEquations(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquations(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:174}{}\section* {$\rm SubtractEquationsLeft(Less)(R)$}

$\rm
[SubtractEquationsLeft(Less)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}SubtractEquationsLeft(Less)(R)
\mbox{''}
]$

$\rm
[SubtractEquationsLeft(Less)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}subtractEquationsLeft(Less)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:175}{}\section* {$\rm LessNegated(Negative)(R)$}

$\rm
[LessNegated(Negative)(R)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LessNegated(Negative)(R)
\mbox{''}
]$

$\rm
[LessNegated(Negative)(R)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}lessNegated(Negative)(R)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:176}{}\section* {$\rm FromNegatedAnd(Imply)$}

$\rm
[FromNegatedAnd(Imply)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash FromNegatedAnd
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\rhd \underline{
\mathsf {a}}
\gg \dot{\neg}\,(\underline{
\mathsf {b}}
)n
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[FromNegatedAnd(Imply)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\underline{
\mathsf {b}}
)n
]$

$\rm
[FromNegatedAnd(Imply)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNegatedAnd(Imply)
\mbox{''}
]$

$\rm
[FromNegatedAnd(Imply)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
\linebreak [0]\ \hskip0em plus2.0em{}negated
\linebreak [0]\ \hskip0em plus2.0em{}and
\linebreak [0]\ \hskip0em plus2.0em{}(imply)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:177}{}\section* {$\rm RemoveDoubleNeg(Consequent)$}

$\rm
[RemoveDoubleNeg(Consequent)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\rhd \underline{
\mathsf {a}}
\gg \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
; RemoveDoubleNeg
\rhd \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\gg \underline{
\mathsf {b}}
; 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon Ded
\rhd 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \underline{
\mathsf {a}}
\vdash \underline{
\mathsf {b}}
\gg \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
; \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash MP
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\Rightarrow \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\rhd \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\gg \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[RemoveDoubleNeg(Consequent)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon \underline{
\mathsf {a}}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
]$

$\rm
[RemoveDoubleNeg(Consequent)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}RemoveDoubleNeg(Consequent)
\mbox{''}
]$

$\rm
[RemoveDoubleNeg(Consequent)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}prop
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}remove
\linebreak [0]\ \hskip0em plus2.0em{}double
\linebreak [0]\ \hskip0em plus2.0em{}neg
\linebreak [0]\ \hskip0em plus2.0em{}(consequent)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:178}{}\section* {$\rm FromNotUpperBound$}

$\rm
[FromNotUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}FromNotUpperBound
\mbox{''}
]$

$\rm
[FromNotUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}fromNotUpperBound
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:179}{}\section* {$\rm LeqNUB$}

$\rm
[LeqNUB
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}LeqNUB
\mbox{''}
]$

$\rm
[LeqNUB
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}leqNUB
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:180}{}\section* {$\rm USlimitIsLeastUpperBound(Helper)$}

$\rm
[USlimitIsLeastUpperBound(Helper)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USlimitIsLeastUpperBound(Helper)
\mbox{''}
]$

$\rm
[USlimitIsLeastUpperBound(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsLeastUpperBound
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:181}{}\section* {$\rm USlimitIsLeastUpperBound$}

$\rm
[USlimitIsLeastUpperBound
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}USlimitIsLeastUpperBound
\mbox{''}
]$

$\rm
[USlimitIsLeastUpperBound
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}USlimitIsLeastUpperBound
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:182}{}\section* {$\rm ExistMP3$}

$\rm
[ExistMP3
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{(v2)}
\colon 
\forall \underline{(v3)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\vdash ExistMP2
\rhd \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\gg \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
; ExistMP
\rhd \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\rhd \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\gg \underline{
\mathsf {d}}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[ExistMP3
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(v1)}
\colon 
\forall \underline{(v2)}
\colon 
\forall \underline{(v3)}
\colon 
\forall \underline{
\mathsf {a}}
\colon 
\forall \underline{
\mathsf {b}}
\colon 
\forall \underline{
\mathsf {c}}
\colon 
\forall \underline{
\mathsf {d}}
\colon \underline{
\mathsf {a}}
\Rightarrow \underline{
\mathsf {b}}
\Rightarrow \underline{
\mathsf {c}}
\Rightarrow \underline{
\mathsf {d}}
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v1)}
\colon \dot{\neg}\,(\underline{
\mathsf {a}}
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v2)}
\colon \dot{\neg}\,(\underline{
\mathsf {b}}
)n
)n
\vdash \dot{\neg}\,(
\forall_{obj} \underline{(v3)}
\colon \dot{\neg}\,(\underline{
\mathsf {c}}
)n
)n
\vdash \underline{
\mathsf {d}}
]$

$\rm
[ExistMP3
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ExistMP3
\mbox{''}
]$

$\rm
[ExistMP3
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}pred
\linebreak [0]\ \hskip0em plus2.0em{}lemma
\linebreak [0]\ \hskip0em plus2.0em{}exist
\linebreak [0]\ \hskip0em plus2.0em{}mp3
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:183}{}\section* {$\rm GreaterPositive(N)$}

$\rm
[GreaterPositive(N)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}GreaterPositive(N)
\mbox{''}
]$

$\rm
[GreaterPositive(N)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}greaterPositive(N)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:184}{}\section* {$\rm ysFClose(Helper)$}

$\rm
[ysFClose(Helper)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysFClose(Helper)
\mbox{''}
]$

$\rm
[ysFClose(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFClose
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:185}{}\section* {$\rm ysFClose$}

$\rm
[ysFClose
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysFClose
\mbox{''}
]$

$\rm
[ysFClose
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFClose
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:186}{}\section* {$\rm ysFCauchy(Helper)$}

$\rm
[ysFCauchy(Helper)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysFCauchy(Helper)
\mbox{''}
]$

$\rm
[ysFCauchy(Helper)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFCauchy
\linebreak [0]\ \hskip0em plus2.0em{}helper
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:187}{}\section* {$\rm ysFCauchy$}

$\rm
[ysFCauchy
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}ysFCauchy
\mbox{''}
]$

$\rm
[ysFCauchy
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}ysFCauchy
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:188}{}\section* {$\rm from<<==$}

$\rm
[from<<==
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash WeakenOr2
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash From==
\gg 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; WeakenOr1
\rhd 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; FromDisjuncts
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[from<<==
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
]$

$\rm
[from<<==
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}from
{<}
{<}
{=}
{=}
\mbox{''}
]$

$\rm
[from<<==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}from
{<}
{<}
{=}
{=}
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:189}{}\section* {$\rm to<<==$}

$\rm
[to<<==
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash Repetition
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
; WeakenOr2
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash To==
\rhd 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; WeakenOr1
\rhd  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Ded
\rhd 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; FromDisjuncts
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\rhd \dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rhd 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
; Repetition
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[to<<==
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon 
\forall \underline{(fy)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fy)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \underline{(fx)}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
 =  \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P( \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
)
\mid \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(r1)}
\colon \overline{(r1)}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(\overline{(r1)}
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(f1)}
\colon 
\forall_{obj} \overline{(f2)}
\colon 
\forall_{obj} \overline{(f3)}
\colon 
\forall_{obj} \overline{(f4)}
\colon \{\{\overline{(f1)}
,\overline{(f1)}
\}
,\{\overline{(f1)}
,\overline{(f2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \{\{\overline{(f3)}
,\overline{(f3)}
\}
,\{\overline{(f3)}
,\overline{(f4)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
\Rightarrow \overline{(f1)}
 = \overline{(f3)}
\Rightarrow \overline{(f2)}
 = \overline{(f4)}
)n
)n
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s1)}
\colon \overline{(s1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(
\forall_{obj} \overline{(s2)}
\colon \dot{\neg}\,(\{\{\overline{(s1)}
,\overline{(s1)}
\}
,\{\overline{(s1)}
,\overline{(s2)}
\}
\}
 \mathrel{\in} 
\mathsf {f}
 _{Ph} 
)n
)n
)n
)n
\}
)
\mid 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|(\underline{(fy)}
[\overline{
\mathsf {m}}
]
+(-u
\mathsf {d}
 _{Ph} 
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\}
]$

$\rm
[to<<==
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}to
{<}
{<}
{=}
{=}
\mbox{''}
]$

$\rm
[to<<==
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}to
{<}
{<}
{=}
{=}
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:190}{}\section* {$\rm NonnegativeNumerical(F)$}

$\rm
[NonnegativeNumerical(F)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash NumericalF
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 = \underline{(fx)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
)n
)n
; FirstConjunct
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 = \underline{(fx)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 = \underline{(fx)}
; MP
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 = \underline{(fx)}
\rhd \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\gg |f\underline{(fx)}
|
 = \underline{(fx)}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NonnegativeNumerical(F)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\vdash |f\underline{(fx)}
|
 = \underline{(fx)}
]$

$\rm
[NonnegativeNumerical(F)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NonnegativeNumerical(F)
\mbox{''}
]$

$\rm
[NonnegativeNumerical(F)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}nonnegativeNumerical(F)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:191}{}\section* {$\rm NegativeNumerical(F)$}

$\rm
[NegativeNumerical(F)
\stackrel{
proof
}{\rightarrow}
\lambda 
\mathsf {c}
.
\lambda 
\mathsf{x}
.
{\cal P}( 
\lceil SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\vdash NumericalF
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 = \underline{(fx)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
)n
)n
; SecondConjunct
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 = \underline{(fx)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
)n
)n
\gg \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
; MP
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\Rightarrow |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
\rhd \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\gg |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
\rceil 
, 
p_0
, 
\mathsf {c}
)
]$

$\rm
[NegativeNumerical(F)
\stackrel{
stmt
}{\rightarrow}SystemQ
\vdash 
\forall \underline{(fx)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(\dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \dot{\neg}\,(\overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
 <= (\underline{(fx)}
[\overline{
\mathsf {m}}
]
+(-u\overline{(\epsilon)}
)
)
)n
)n
)n
)n
)n
)n
)n
\Rightarrow 
\forall_{obj} \overline{(\epsilon)}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {n}}
\colon \dot{\neg}\,(
\forall_{obj} \overline{
\mathsf {m}}
\colon \dot{\neg}\,(0
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(0
 = \overline{(\epsilon)}
)n
)n
)n
\Rightarrow \overline{
\mathsf {n}}
 <= \overline{
\mathsf {m}}
\Rightarrow \dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 <= \overline{(\epsilon)}
\Rightarrow \dot{\neg}\,(\dot{\neg}\,(|( \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \overline{(crs1)}
\colon \dot{\neg}\,(
\mathsf {c}
 _{Ph} 
 = \{\{\overline{(crs1)}
,\overline{(crs1)}
\}
,\{\overline{(crs1)}
,0
\}
\}
)n
)n
\}
[\overline{
\mathsf {m}}
]
+(-u\underline{(fx)}
[\overline{
\mathsf {m}}
]
)
)
|
 = \overline{(\epsilon)}
)n
)n
)n
)n
)n
)n
\vdash |f\underline{(fx)}
|
 =  \{ ph \mathrel{\in}  \{ ph \mathrel{\in} P(P(Union(\{N
,Q
\}
)
)
)
\mid \dot{\neg}\,(
\forall_{obj} \overline{(op1)}
\colon \dot{\neg}\,(\dot{\neg}\,(
\forall_{obj} \overline{(op2)}
\colon \dot{\neg}\,(\dot{\neg}\,(\dot{\neg}\,(\overline{(op1)}
 \mathrel{\in} N
\Rightarrow \dot{\neg}\,(\overline{(op2)}
 \mathrel{\in} Q
)n
)n
\Rightarrow \dot{\neg}\,(
\mathsf {a}
 _{Ph} 
 = \{\{\overline{(op1)}
,\overline{(op1)}
\}
,\{\overline{(op1)}
,\overline{(op2)}
\}
\}
)n
)n
)n
)n
)n
)n
\}
\mid \dot{\neg}\,(
\forall_{obj} \underline{
\mathsf {m}}
\colon \dot{\neg}\,(
\mathsf {f}
 _{Ph} 
 = \{\{\underline{
\mathsf {m}}
,\underline{
\mathsf {m}}
\}
,\{\underline{
\mathsf {m}}
,(-u\underline{(fx)}
[\underline{
\mathsf {m}}
]
)
\}
\}
)n
)n
\}
]$

$\rm
[NegativeNumerical(F)
\stackrel{
\mathrm{tex}
}{\rightarrow}
\mbox{``}NegativeNumerical(F)
\mbox{''}
]$

$\rm
[NegativeNumerical(F)
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}lemma
\linebreak [0]\ \hskip0em plus2.0em{}negativeNumerical(F)
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:192}{}\section* {$\rm 
\mbox{tester1}$}

$\rm
[
\mbox{tester1}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester1
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:193}{}\section* {$\rm 
\mbox{tester2}$}

$\rm
[
\mbox{tester2}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester2
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:194}{}\section* {$\rm 
\mbox{tester3}$}

$\rm
[
\mbox{tester3}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester3
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:195}{}\section* {$\rm 
\mbox{tester4}$}

$\rm
[
\mbox{tester4}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester4
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:196}{}\section* {$\rm 
\mbox{tester5}$}

$\rm
[
\mbox{tester5}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester5
\mbox{''}
]$

\hypertarget {url:1808490797518007916660448878599179667361805706605856764796691709371934721:197}{}\section* {$\rm 
\mbox{tester6}$}

$\rm
[
\mbox{tester6}
\stackrel{
\mathrm{pyk}
}{\rightarrow}
\mbox{``}tester6
\mbox{''}
]$

\vspace{2ex}

{\em \href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/compiler/index.html}{The pyk compiler},
version 0.grue.20060417+
by
\href {http://www.diku.dk/~grue/index.html}{Klaus Grue}\\
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{GRD-2006-12-29.UTC:10:12:14.905583} = 
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{MJD-54098.TAI:10:12:47.905583} = 
\href {http://www.diku.dk/~grue/logiweb/20060417/logiweb/doc/misc/time.html}{LGT-4674103967905583e-6}
}

\end{document}

