Logiweb aspects of lemma splitNumericalSum(+-) in pyk

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The predefined "pyk" aspect

define pyk of lemma splitNumericalSum(+-) 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 right parenthesis unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma splitNumericalSum(+-) 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 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(+-) 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 | 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 splitNumericalSum(+-) 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 | metavar var y end metavar | <= | metavar var x end metavar | infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens | metavar var y end metavar | <= | metavar var x end metavar | conclude | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | cut lemma 0<=|x| conclude 0 <= | metavar var y end metavar | cut lemma leqAdditionLeft modus ponens 0 <= | metavar var y end metavar | conclude | metavar var x end metavar | + 0 <= | metavar var x end metavar | + | metavar var y end metavar | cut axiom plus0 conclude | metavar var x end metavar | + 0 = | metavar var x end metavar | cut lemma subLeqLeft modus ponens | metavar var x end metavar | + 0 = | metavar var x end metavar | modus ponens | metavar var x end metavar | + 0 <= | metavar var x end metavar | + | metavar var y end metavar | conclude | metavar var x end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma leqTransitivity modus ponens | metavar var x end metavar + metavar var y end metavar | <= | metavar var x end metavar | modus ponens | 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 not0 | metavar var y end metavar | <= | metavar var x end metavar | infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma toLess modus ponens not0 | metavar var y end metavar | <= | metavar var x 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 splitNumericalSum(+-, bigNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 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 | <= | metavar var y end metavar | cut lemma 0<=|x| conclude 0 <= | metavar var x end metavar | cut lemma leqAddition modus ponens 0 <= | metavar var x end metavar | conclude 0 + | metavar var y end metavar | <= | metavar var x end metavar | + | metavar var y end metavar | cut lemma plus0Left conclude 0 + | metavar var y end metavar | = | metavar var y end metavar | cut lemma subLeqLeft modus ponens 0 + | metavar var y end metavar | = | metavar var y end metavar | modus ponens 0 + | metavar var y end metavar | <= | 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 leqTransitivity modus ponens | metavar var x end metavar + metavar var y 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 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 | metavar var y end metavar | <= | metavar var x end metavar | infer 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 | metavar var y end metavar | <= | metavar var x end metavar | imply 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 not0 | metavar var y end metavar | <= | metavar var x end metavar | infer 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 not0 | metavar var y end metavar | <= | metavar var x end metavar | imply 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 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer prop lemma from negations modus ponens | metavar var y end metavar | <= | metavar var x end metavar | imply 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 not0 | metavar var y end metavar | <= | metavar var x end metavar | imply 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 | 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 prop lemma mp2 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 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 | end quote state proof state cache var c end expand end define