Logiweb aspects of lemma fromNotSameF(Weak)(Helper) in pyk

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The predefined "pyk" aspect

define pyk of lemma fromNotSameF(Weak)(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 f unicode small r unicode small o unicode small m unicode capital n unicode small o unicode small t unicode capital s unicode small a unicode small m unicode small e unicode capital f unicode left parenthesis unicode capital w unicode small e unicode small a unicode small k unicode right parenthesis 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 predefined "tex" aspect

define tex of lemma fromNotSameF(Weak)(Helper) as text unicode start of text unicode capital f unicode small r unicode small o unicode small m unicode capital n unicode small o unicode small t unicode capital s unicode small a unicode small m unicode small e unicode capital f unicode left parenthesis unicode capital w unicode small e unicode small a unicode small k unicode right parenthesis 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 fromNotSameF(Weak)(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 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 infer not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma fromNotSameF(Weak)(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 metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer 0 <= metavar var x end metavar + - metavar var y end metavar infer lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar conclude | metavar var x end metavar + - metavar var y end metavar | = metavar var x end metavar + - metavar var y end metavar cut lemma subLeqRight 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 z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude metavar var z end metavar <= metavar var x end metavar + - metavar var y end metavar cut lemma negativeToLeft(Leq) modus ponens metavar var z end metavar <= metavar var x end metavar + - metavar var y end metavar conclude metavar var z end metavar + metavar var y end metavar <= metavar var x 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 subLeqLeft modus ponens metavar var z end metavar + metavar var y end metavar = metavar var y end metavar + metavar var z end metavar modus ponens metavar var z end metavar + metavar var y end metavar <= metavar var x end metavar conclude metavar var y end metavar + metavar var z end metavar <= metavar var x end metavar cut lemma positiveToRight(Leq) modus ponens metavar var y end metavar + metavar var z end metavar <= metavar var x end metavar conclude metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut prop lemma weaken or first modus ponens metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer not0 0 <= metavar var x end metavar + - metavar var y end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar conclude not0 metavar var x end metavar + - metavar var y end metavar <= 0 imply not0 not0 metavar var x end metavar + - metavar var y end metavar = 0 cut lemma negativeNumerical conclude | metavar var x end metavar + - metavar var y end metavar | = - metavar var x end metavar + - metavar var y end metavar cut lemma minusNegated conclude - metavar var x end metavar + - metavar var y end metavar = metavar var y end metavar + - metavar var x end metavar cut lemma eqTransitivity 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 subLeqRight modus ponens | metavar var x end metavar + - metavar var y end metavar | = metavar var y end metavar + - metavar var x end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude metavar var z end metavar <= metavar var y end metavar + - metavar var x end metavar cut lemma negativeToLeft(Leq) modus ponens metavar var z end metavar <= metavar var y end metavar + - metavar var x end metavar conclude metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar cut axiom plusCommutativity conclude metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar cut lemma subLeqLeft modus ponens metavar var z end metavar + metavar var x end metavar = metavar var x end metavar + metavar var z end metavar modus ponens metavar var z end metavar + metavar var x end metavar <= metavar var y end metavar conclude metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar cut lemma positiveToRight(Leq) modus ponens metavar var x end metavar + metavar var z end metavar <= metavar var y end metavar conclude metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar cut prop lemma weaken or second modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer 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 metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | infer not0 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 metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut 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 infer lemma fromNotLess modus ponens 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 conclude metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | cut 1rule mp modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut 1rule mp modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | imply not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar modus ponens metavar var z end metavar <= | metavar var x end metavar + - metavar var y end metavar | conclude not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar cut prop lemma from negations modus ponens 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar modus ponens not0 0 <= metavar var x end metavar + - metavar var y end metavar imply not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar conclude not0 metavar var x end metavar <= metavar var y end metavar + - metavar var z end metavar imply metavar var y end metavar <= metavar var x end metavar + - metavar var z end metavar end quote state proof state cache var c end expand end define