Logiweb aspects of lemma splitNumericalSum(+-, smallNegative) in pyk

Up Help

The predefined "pyk" aspect

define pyk of lemma splitNumericalSum(+-, smallNegative) as text unicode start of text unicode small l unicode small e unicode small m unicode small m unicode small a unicode space unicode small s unicode small p unicode small l unicode small i unicode small t unicode capital n unicode small u unicode small m unicode small e unicode small r unicode small i unicode small c unicode small a unicode small l unicode capital s unicode small u unicode small m unicode left parenthesis unicode plus sign unicode hyphen unicode comma unicode space unicode small s unicode small m unicode small a unicode small l unicode small l unicode capital n unicode small e unicode small g unicode small a unicode small t unicode small i unicode small v unicode small e unicode right parenthesis unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma splitNumericalSum(+-, smallNegative) as text unicode start of text unicode small s unicode small p unicode small l unicode small i unicode small t unicode capital n unicode small u unicode small m unicode small e unicode small r unicode small i unicode small c unicode small a unicode small l unicode capital s unicode small u unicode small m unicode left parenthesis unicode plus sign unicode hyphen unicode small s unicode small m unicode small a unicode small l unicode small l unicode right parenthesis unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma splitNumericalSum(+-, smallNegative) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) end define

The user defined "the proof aspect" aspect

define proof of lemma splitNumericalSum(+-, smallNegative) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer lemma leqAdditionLeft modus ponens metavar var y end metavar <= 0 conclude metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar + 0 cut axiom plus0 conclude metavar var x end metavar + 0 = metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar + 0 = metavar var x end metavar modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar + 0 conclude metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar cut lemma positiveToRight(Leq)(1 term) modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma nonpositiveNumerical modus ponens metavar var y end metavar <= 0 conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - metavar var y end metavar cut lemma eqNegated modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - metavar var y end metavar conclude - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - - metavar var y end metavar cut lemma doubleMinus conclude - - metavar var y end metavar = metavar var y end metavar cut lemma eqTransitivity modus ponens - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = - - metavar var y end metavar modus ponens - - metavar var y end metavar = metavar var y end metavar conclude - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma addEquations modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var y end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar cut lemma subLeqRight modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar modus ponens 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + - if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar + metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = metavar var x end metavar + metavar var y end metavar conclude metavar var x end metavar + metavar var y end metavar = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar conclude metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var y end metavar = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) modus ponens metavar var x end metavar + metavar var y end metavar <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= metavar var x end metavar cut lemma subLeqRight modus ponens metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) end quote state proof state cache var c end expand end define