Logiweb aspects of lemma toCartProd helper in pyk

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The predefined "pyk" aspect

define pyk of lemma toCartProd helper 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 t unicode small o unicode capital c unicode small a unicode small r unicode small t unicode capital p unicode small r unicode small o unicode small d unicode space unicode small h unicode small e unicode small l unicode small p unicode small e unicode small r unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma toCartProd helper as text unicode start of text unicode capital t unicode small o unicode capital c unicode small a unicode small r unicode small t unicode capital p unicode small r unicode small o unicode small d unicode left parenthesis unicode capital h unicode small e unicode small l unicode small p unicode small e unicode small r 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 toCartProd helper as system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end define

The user defined "the proof aspect" aspect

define proof of lemma toCartProd helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer object var var s1 end var in0 metavar var sz end metavar infer lemma fromOrderedPair(twoLevels) modus ponens object var var s1 end var in0 metavar var sz end metavar modus ponens metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude not0 object var var s1 end var = metavar var sx end metavar imply object var var s1 end var = metavar var sy end metavar cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer object var var s1 end var = metavar var sx end metavar infer lemma sameMember(2) modus ponens object var var s1 end var = metavar var sx end metavar modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar conclude object var var s1 end var in0 metavar var sx1 end metavar cut lemma toBinaryUnion(1) modus ponens object var var s1 end var in0 metavar var sx1 end metavar conclude object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sy end metavar in0 metavar var sy1 end metavar infer object var var s1 end var = metavar var sy end metavar infer lemma sameMember(2) modus ponens object var var s1 end var = metavar var sy end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude object var var s1 end var in0 metavar var sy1 end metavar cut lemma toBinaryUnion(2) modus ponens object var var s1 end var in0 metavar var sy1 end metavar conclude object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy1 end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer object var var s1 end var = metavar var sx end metavar infer object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude metavar var sx end metavar in0 metavar var sx1 end metavar imply object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule mp modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar imply object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar conclude object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule deduction modus ponens all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed metavar var sy end metavar in0 metavar var sy1 end metavar infer object var var s1 end var = metavar var sy end metavar infer object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude metavar var sy end metavar in0 metavar var sy1 end metavar imply object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule mp modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar imply object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar conclude object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut prop lemma from disjuncts modus ponens not0 object var var s1 end var = metavar var sx end metavar imply object var var s1 end var = metavar var sy end metavar modus ponens object var var s1 end var = metavar var sx end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens object var var s1 end var = metavar var sy end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed 1rule deduction modus ponens all metavar var sx end metavar indeed all metavar var sx1 end metavar indeed all metavar var sy end metavar indeed all metavar var sy1 end metavar indeed all metavar var sz end metavar indeed metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer object var var s1 end var in0 metavar var sz end metavar infer object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude metavar var sx end metavar in0 metavar var sx1 end metavar imply metavar var sy end metavar in0 metavar var sy1 end metavar imply metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut metavar var sx end metavar in0 metavar var sx1 end metavar infer metavar var sy end metavar in0 metavar var sy1 end metavar infer metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair infer prop lemma mp3 modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar imply metavar var sy end metavar in0 metavar var sy1 end metavar imply metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair imply object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) modus ponens metavar var sx end metavar in0 metavar var sx1 end metavar modus ponens metavar var sy end metavar in0 metavar var sy1 end metavar modus ponens metavar var sz end metavar in0 zermelo pair zermelo pair metavar var sx end metavar comma metavar var sx end metavar end pair comma zermelo pair metavar var sx end metavar comma metavar var sy end metavar end pair end pair conclude object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule gen modus ponens object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) cut 1rule repetition modus ponens for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) conclude for all objects object var var s1 end var indeed object var var s1 end var in0 metavar var sz end metavar imply object var var s1 end var in0 U( zermelo pair metavar var sx1 end metavar comma metavar var sy1 end metavar end pair ) end quote state proof state cache var c end expand end define