Logiweb aspects of lemma splitNumericalProduct in pyk

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The predefined "pyk" aspect

define pyk of lemma splitNumericalProduct 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 s unicode small p unicode small l unicode small i unicode small t 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 p unicode small r unicode small o unicode small d unicode small u unicode small c unicode small t unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma splitNumericalProduct as text unicode start of text unicode capital s unicode small p unicode small l unicode small i unicode small t 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 p unicode small r unicode small o unicode small d unicode small u unicode small c unicode small t unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma splitNumericalProduct as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | end define

The user defined "the proof aspect" aspect

define proof of lemma splitNumericalProduct 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 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma splitNumericalProduct(++) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= 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 all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalProduct(+-) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 conclude | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma splitNumericalProduct(+-) modus ponens 0 <= metavar var y end metavar modus ponens metavar var x end metavar <= 0 conclude | metavar var y end metavar * metavar var x end metavar | = | metavar var y end metavar | * | metavar var x end metavar | cut axiom timesCommutativity conclude metavar var x end metavar * metavar var y end metavar = metavar var y end metavar * metavar var x end metavar cut lemma sameNumerical 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 axiom timesCommutativity conclude | metavar var y end metavar | * | metavar var x end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma eqTransitivity4 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 y end metavar * metavar var x end metavar | = | metavar var y end metavar | * | metavar var x end metavar | modus ponens | metavar var y end metavar | * | metavar var x end metavar | = | 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 all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma nonpositiveNegated modus ponens metavar var x end metavar <= 0 conclude 0 <= - metavar var x end metavar cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma splitNumericalProduct(++) modus ponens 0 <= - metavar var x end metavar modus ponens 0 <= - 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 minusTimesMinus conclude - metavar var x end metavar * - metavar var y end metavar = metavar var x end metavar * metavar var y end metavar cut lemma sameNumerical modus ponens - metavar var x end metavar * - metavar var y end metavar = 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 eqSymmetry modus ponens | - metavar var x end metavar * - metavar var y end metavar | = | 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 signNumerical conclude | metavar var x end metavar | = | - metavar var x end metavar | cut lemma signNumerical conclude | metavar var y end metavar | = | - metavar var y end metavar | cut lemma multiplyEquations modus ponens | metavar var x end metavar | = | - metavar var x end metavar | modus ponens | metavar var y 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 eqSymmetry modus ponens | metavar var x end metavar | * | metavar var y end metavar | = | - 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 eqTransitivity4 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 x end metavar | * | - metavar var y end metavar | modus ponens | - metavar var x end metavar | * | - metavar var y end metavar | = | 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 all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude metavar var x end metavar <= 0 imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar imply | 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 <= 0 imply 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply | 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 <= 0 imply metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | conclude metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | cut lemma from leqGeq modus ponens 0 <= metavar var y end metavar imply | metavar var x end metavar * metavar var y end metavar | = | metavar var x end metavar | * | metavar var y end metavar | modus ponens metavar var y end metavar <= 0 imply | metavar var x end metavar * metavar var y end metavar | = | 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 | end quote state proof state cache var c end expand end define