Logiweb aspects of lemma splitNumericalSum(+-, bigNegative) in pyk

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The predefined "pyk" aspect

define pyk of lemma splitNumericalSum(+-, bigNegative) 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 s unicode small u unicode small m unicode left parenthesis unicode plus sign unicode hyphen unicode comma unicode space unicode small b unicode small i unicode small g unicode capital n unicode small e unicode small g unicode small a unicode small t unicode small i unicode small v unicode small e unicode right parenthesis unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma splitNumericalSum(+-, bigNegative) as text unicode start of text 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 s unicode small u unicode small m unicode left parenthesis unicode plus sign unicode hyphen unicode small b unicode small i unicode small g 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 splitNumericalSum(+-, bigNegative) as system Q infer 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 not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | infer | metavar var x end metavar + metavar var y end metavar | <= | metavar var y end metavar | end define

The user defined "the proof aspect" aspect

define proof of lemma splitNumericalSum(+-, bigNegative) 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 metavar var y end metavar <= 0 infer not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | infer lemma nonnegativeNegated modus ponens 0 <= metavar var x end metavar conclude - metavar var x end metavar <= 0 cut lemma nonpositiveNegated modus ponens metavar var y end metavar <= 0 conclude 0 <= - metavar var y end metavar cut lemma signNumerical conclude | metavar var x end metavar | = | - metavar var x end metavar | cut lemma subLessLeft modus ponens | metavar var x end metavar | = | - metavar var x end metavar | modus ponens not0 | metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | metavar var x end metavar | = | metavar var y end metavar | conclude not0 | - metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | metavar var y end metavar | cut lemma signNumerical conclude | metavar var y end metavar | = | - metavar var y end metavar | cut lemma subLessRight modus ponens | metavar var y end metavar | = | - metavar var y end metavar | modus ponens not0 | - metavar var x end metavar | <= | metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | metavar var y end metavar | conclude not0 | - metavar var x end metavar | <= | - metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | - metavar var y end metavar | cut lemma lessLeq modus ponens not0 | - metavar var x end metavar | <= | - metavar var y end metavar | imply not0 not0 | - metavar var x end metavar | = | - metavar var y end metavar | conclude | - metavar var x end metavar | <= | - metavar var y end metavar | cut lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= - metavar var y end metavar modus ponens - metavar var x end metavar <= 0 modus ponens | - metavar var x end metavar | <= | - metavar var y end metavar | conclude | - 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 y end metavar | = | - metavar var x end metavar + metavar var y end metavar | cut lemma -x-y=-(x+y) conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var x end metavar + metavar var y end metavar cut axiom plusCommutativity conclude - metavar var x end metavar + - metavar var y end metavar = - metavar var y end metavar + - metavar var x end metavar cut lemma equality 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 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 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 eqSymmetry 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 y end metavar + - metavar var x end metavar | = | metavar var x end metavar + metavar var y end metavar | cut lemma eqSymmetry modus ponens | metavar var y end metavar | = | - metavar var y end metavar | conclude | - metavar var y end metavar | = | metavar var y end metavar | cut lemma subLeqLeft modus ponens | - metavar var y end metavar + - metavar var x end metavar | = | metavar var x end metavar + metavar var y end metavar | modus ponens | - 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 y end metavar | cut lemma subLeqRight modus ponens | - metavar var y end metavar | = | metavar var y end metavar | modus ponens | metavar var x end metavar + metavar var y end metavar | <= | - metavar var y end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var y end metavar | end quote state proof state cache var c end expand end define