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
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
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
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
The pyk compiler, version 0.grue.20060417+ by Klaus Grue,