Logiweb aspects of lemma numericalDifferenceLess helper in pyk

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The predefined "pyk" aspect

define pyk of lemma numericalDifferenceLess helper 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 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 d unicode small i unicode small f unicode small f unicode small e unicode small r unicode small e unicode small n unicode small c unicode small e unicode capital l unicode small e unicode small s unicode small s unicode space unicode small h unicode small e unicode small l unicode small p unicode small e unicode small r unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma numericalDifferenceLess helper as text unicode start of text 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 d unicode small i unicode small f unicode small f unicode small e unicode small r unicode small e unicode small n unicode small c unicode small e unicode capital l unicode small e unicode small s unicode small s unicode left parenthesis unicode capital h unicode small e unicode small l unicode small p unicode small e unicode small r 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 numericalDifferenceLess helper as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar infer not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma numericalDifferenceLess helper 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 all metavar var z end metavar indeed 0 <= metavar var x end metavar + - metavar var y end metavar infer not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar infer lemma leqLessTransitivity modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar conclude not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar cut lemma positiveNegated modus ponens not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar conclude not0 - metavar var z end metavar <= 0 imply not0 not0 - metavar var z end metavar = 0 cut lemma lessAdditionLeft modus ponens not0 - metavar var z end metavar <= 0 imply not0 not0 - metavar var z end metavar = 0 conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar + 0 imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar + 0 cut axiom plus0 conclude metavar var y end metavar + 0 = metavar var y end metavar cut lemma subLessRight modus ponens metavar var y end metavar + 0 = metavar var y end metavar modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar + 0 imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar + 0 conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar cut lemma negativeToLeft(Leq)(1 term) modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar conclude metavar var y end metavar <= metavar var x end metavar cut lemma lessLeqTransitivity modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var y end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var y end metavar modus ponens metavar var y end metavar <= metavar var x end metavar conclude not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar cut lemma negativeToRight(Less) modus ponens not0 metavar var x end metavar + - metavar var y end metavar <= metavar var z end metavar imply not0 not0 metavar var x end metavar + - metavar var y end metavar = metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + metavar var y end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar cut lemma subLessRight modus ponens metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar modus ponens not0 metavar var x end metavar <= metavar var z end metavar + metavar var y end metavar imply not0 not0 metavar var x end metavar = metavar var z end metavar + metavar var y end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar cut prop lemma join conjuncts modus ponens not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar modus ponens not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar conclude not0 not0 metavar var y end metavar + - metavar var z end metavar <= metavar var x end metavar imply not0 not0 metavar var y end metavar + - metavar var z end metavar = metavar var x end metavar imply not0 not0 metavar var x end metavar <= metavar var y end metavar + metavar var z end metavar imply not0 not0 metavar var x end metavar = metavar var y end metavar + metavar var z end metavar end quote state proof state cache var c end expand end define