define pyk of lemma lessMultiplication(F) helper2 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 l unicode small e unicode small s unicode small s unicode capital m unicode small u unicode small l unicode small t unicode small i unicode small p unicode small l unicode small i unicode small c unicode small a unicode small t unicode small i unicode small o unicode small n unicode left parenthesis unicode capital f unicode right parenthesis unicode space unicode small h unicode small e unicode small l unicode small p unicode small e unicode small r unicode two unicode end of text end unicode text end text end define
define tex of lemma lessMultiplication(F) helper2 as text unicode start of text unicode capital l unicode small e unicode small s unicode small s unicode capital m unicode small u unicode small l unicode small t unicode small i unicode small p unicode small l unicode small i unicode small c unicode small a unicode small t unicode small i unicode small o unicode small n unicode left parenthesis unicode capital f 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 two unicode right parenthesis unicode end of text end unicode text end text end define
define statement of lemma lessMultiplication(F) helper2 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 all metavar var u end metavar indeed all metavar var v end metavar indeed not0 0 <= metavar var u end metavar imply not0 not0 0 = metavar var u end metavar infer not0 0 <= metavar var v end metavar imply not0 not0 0 = metavar var v end metavar infer metavar var x end metavar <= metavar var y end metavar + - metavar var u end metavar infer 0 <= metavar var z end metavar + - metavar var v end metavar infer metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var u end metavar * metavar var v end metavar end define
define proof of lemma lessMultiplication(F) helper2 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 all metavar var u end metavar indeed all metavar var v end metavar indeed not0 0 <= metavar var u end metavar imply not0 not0 0 = metavar var u end metavar infer not0 0 <= metavar var v end metavar imply not0 not0 0 = metavar var v end metavar infer metavar var x end metavar <= metavar var y end metavar + - metavar var u end metavar infer 0 <= metavar var z end metavar + - metavar var v end metavar infer lemma negativeToLeft(Leq) modus ponens metavar var x end metavar <= metavar var y end metavar + - metavar var u end metavar conclude metavar var x end metavar + metavar var u end metavar <= metavar var y end metavar cut axiom plusCommutativity conclude metavar var x end metavar + metavar var u end metavar = metavar var u end metavar + metavar var x end metavar cut lemma subLeqLeft modus ponens metavar var x end metavar + metavar var u end metavar = metavar var u end metavar + metavar var x end metavar modus ponens metavar var x end metavar + metavar var u end metavar <= metavar var y end metavar conclude metavar var u end metavar + metavar var x end metavar <= metavar var y end metavar cut lemma positiveToRight(Leq) modus ponens metavar var u end metavar + metavar var x end metavar <= metavar var y end metavar conclude metavar var u end metavar <= metavar var y end metavar + - metavar var x end metavar cut lemma negativeToLeft(Leq)(1 term) modus ponens 0 <= metavar var z end metavar + - metavar var v end metavar conclude metavar var v end metavar <= metavar var z end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var u end metavar imply not0 not0 0 = metavar var u end metavar conclude 0 <= metavar var u end metavar cut lemma lessLeq modus ponens not0 0 <= metavar var v end metavar imply not0 not0 0 = metavar var v end metavar conclude 0 <= metavar var v end metavar cut lemma multiplyEquations(Leq) modus ponens 0 <= metavar var u end metavar modus ponens 0 <= metavar var v end metavar modus ponens metavar var u end metavar <= metavar var y end metavar + - metavar var x end metavar modus ponens metavar var v end metavar <= metavar var z end metavar conclude metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar cut axiom timesCommutativity conclude metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar cut lemma distributionLeft conclude metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma -x*y=-(x*y) conclude - metavar var x end metavar * metavar var z end metavar = - metavar var x end metavar * metavar var z end metavar cut lemma eqAdditionLeft modus ponens - metavar var x end metavar * metavar var z end metavar = - metavar var x end metavar * metavar var z end metavar conclude metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma eqTransitivity4 modus ponens metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar modus ponens metavar var z end metavar * metavar var y end metavar + - metavar var x end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar modus ponens metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar conclude metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar = metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar modus ponens metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar + - metavar var x end metavar * metavar var z end metavar conclude metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar cut lemma negativeToLeft(Leq) modus ponens metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var x end metavar * metavar var z end metavar conclude metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar cut axiom plusCommutativity conclude metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar cut lemma subLeqLeft modus ponens metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar = metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar modus ponens metavar var u end metavar * metavar var v end metavar + metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar cut lemma positiveToRight(Leq) modus ponens metavar var x end metavar * metavar var z end metavar + metavar var u end metavar * metavar var v end metavar <= metavar var y end metavar * metavar var z end metavar conclude metavar var x end metavar * metavar var z end metavar <= metavar var y end metavar * metavar var z end metavar + - metavar var u end metavar * metavar var v end metavar end quote state proof state cache var c end expand end define
The pyk compiler, version 0.grue.20060417+ by Klaus Grue,