Logiweb aspects of lemma sameNumerical in pyk

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The predefined "pyk" aspect

define pyk of lemma sameNumerical 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 a unicode small m unicode small e 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 sameNumerical as text unicode start of text unicode capital s unicode small a unicode small m unicode small e 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 sameNumerical as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer 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 ) end define

The user defined "the proof aspect" aspect

define proof of lemma sameNumerical 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 x end metavar = metavar var y end metavar infer 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 subLeqRight modus ponens metavar var x end metavar = metavar var y end metavar modus ponens 0 <= metavar var x end metavar conclude 0 <= metavar var y end metavar cut lemma nonnegativeNumerical modus ponens 0 <= 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 eqSymmetry 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 metavar var y end metavar = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma eqTransitivity4 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 metavar var x end metavar = metavar var y end metavar modus ponens 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 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 all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 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 subLessLeft modus ponens metavar var x end metavar = metavar var y end metavar modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 metavar var y end metavar <= 0 imply not0 not0 metavar var y end metavar = 0 cut lemma negativeNumerical modus ponens not0 metavar var y end metavar <= 0 imply not0 not0 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 eqSymmetry 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 - metavar var y end metavar = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma eqNegated modus ponens metavar var x end metavar = metavar var y end metavar conclude - metavar var x end metavar = - metavar var y end metavar cut lemma eqTransitivity4 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 - metavar var x end metavar = - metavar var y end metavar modus ponens - 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 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 all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar = metavar var y end metavar infer 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer 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 imply metavar var x end metavar = metavar var y end metavar imply 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 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 0 <= metavar var x end metavar infer metavar var x end metavar = metavar var y end metavar infer 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 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply 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 prop lemma from negations modus ponens 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply 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 ) modus ponens not0 0 <= metavar var x end metavar imply metavar var x end metavar = metavar var y end metavar imply 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 metavar var x end metavar = metavar var y end metavar imply 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 1rule mp modus ponens metavar var x end metavar = metavar var y end metavar imply 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 ) modus ponens metavar var x 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 ) end quote state proof state cache var c end expand end define