Logiweb aspects of lemma lessMultiplication(F) helper2 in pyk

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The predefined "pyk" aspect

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

The predefined "tex" aspect

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

The user defined "the statement aspect" aspect

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

The user defined "the proof aspect" aspect

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