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

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The predefined "pyk" aspect

define pyk of lemma splitNumericalSum(+-, bigNegative) 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 b unicode small i unicode small g 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(+-, bigNegative) 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 b unicode small i unicode small g 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(+-, bigNegative) 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 not0 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 ) imply not0 not0 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 ) 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 y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define

The user defined "the proof aspect" aspect

define proof of lemma splitNumericalSum(+-, bigNegative) 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 not0 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 ) imply not0 not0 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 ) infer lemma nonnegativeNegated modus ponens 0 <= metavar var x end metavar conclude - metavar var x end metavar <= 0 cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma signNumerical conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) cut lemma subLessLeft modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= - metavar var x end metavar , - metavar var x end metavar , - - metavar var x end metavar ) modus ponens not0 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 ) imply not0 not0 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 not0 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 ) imply not0 not0 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 signNumerical conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma subLessRight modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) modus ponens not0 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 ) imply not0 not0 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 not0 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 ) imply not0 not0 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 lessLeq modus ponens not0 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 ) imply not0 not0 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 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 splitNumericalSum(+-, smallNegative) modus ponens 0 <= - metavar var y end metavar modus ponens - metavar var x end metavar <= 0 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 ) conclude if( 0 <= - 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 ) <= if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma signNumerical 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 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 -x-y=-(x+y) conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut axiom plusCommutativity conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma equality modus ponens - 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 y end metavar + - metavar var x end metavar conclude - metavar var x end metavar + metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar + metavar var y 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 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 eqTransitivity 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 ) = 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 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 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 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 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 ) = if( 0 <= - 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 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 ) = 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 y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= - 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 ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= - 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 ) = 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 if( 0 <= - 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 ) <= if( 0 <= - 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 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 y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) cut lemma subLeqRight modus ponens if( 0 <= - metavar var y end metavar , - metavar var y end metavar , - - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y 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 ) <= if( 0 <= - 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 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 y end metavar , metavar var y end metavar , - metavar var y end metavar ) end quote state proof state cache var c end expand end define