Logiweb aspects of lemma signNumerical in pyk

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The predefined "pyk" aspect

define pyk of lemma signNumerical 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 i unicode small g unicode small n 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 end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma signNumerical as text unicode start of text unicode capital s unicode small i unicode small g unicode small n 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 end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma signNumerical as system Q infer all metavar var x end metavar indeed 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 ) end define

The user defined "the proof aspect" aspect

define proof of lemma signNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma negativeNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma signNumerical(+) modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar 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 doubleMinus conclude - - metavar var x end metavar = metavar var x end metavar cut lemma sameNumerical modus ponens - - metavar var x end metavar = metavar var x end metavar 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 eqTransitivity 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 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 ) 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 eqSymmetry 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 ) 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 all metavar var x end metavar indeed metavar var x end metavar = 0 infer lemma eqNegated modus ponens metavar var x end metavar = 0 conclude - metavar var x end metavar = - 0 cut lemma -0=0 conclude - 0 = 0 cut lemma eqSymmetry modus ponens metavar var x end metavar = 0 conclude 0 = metavar var x end metavar cut lemma eqTransitivity4 modus ponens - metavar var x end metavar = - 0 modus ponens - 0 = 0 modus ponens 0 = metavar var x end metavar conclude - metavar var x end metavar = metavar var x end metavar cut lemma eqSymmetry modus ponens - metavar var x end metavar = metavar var x end metavar conclude metavar var x end metavar = - metavar var x end metavar cut lemma sameNumerical modus ponens metavar var x end metavar = - metavar var x end metavar 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 all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer lemma signNumerical(+) modus ponens not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar 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 all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer 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 ) conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply 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 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar = 0 infer 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 ) conclude metavar var x end metavar = 0 imply 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 1rule deduction modus ponens all metavar var x end metavar indeed not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar infer 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 ) conclude not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply 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 lessTotality conclude not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply not0 metavar var x end metavar = 0 imply not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar cut prop lemma from three disjuncts modus ponens not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply not0 metavar var x end metavar = 0 imply not0 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply 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 metavar var x end metavar = 0 imply 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 0 <= metavar var x end metavar imply not0 not0 0 = metavar var x end metavar imply 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 ) 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 ) end quote state proof state cache var c end expand end define