Logiweb aspects of lemma lessNeq(F) helper in pyk

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The predefined "pyk" aspect

define pyk of lemma lessNeq(F) 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 l unicode small e unicode small s unicode small s unicode capital n unicode small e unicode small q 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 end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma lessNeq(F) helper as text unicode start of text unicode capital l unicode small e unicode small s unicode small s unicode capital n unicode small e unicode small q 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 right parenthesis unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma lessNeq(F) helper as system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma lessNeq(F) helper as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar infer lemma a4 at metavar var n end metavar modus ponens for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar cut prop lemma first conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar conclude not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar cut prop lemma second conjunct modus ponens not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar conclude metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar cut axiom leqReflexivity conclude metavar var n end metavar <= metavar var n end metavar cut 1rule mp modus ponens metavar var n end metavar <= metavar var n end metavar imply [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar modus ponens metavar var n end metavar <= metavar var n end metavar conclude [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar cut lemma switchTerms(x<=y-z) modus ponens [ metavar var fx end metavar ; metavar var n end metavar ] <= [ metavar var fy end metavar ; metavar var n end metavar ] + - metavar var ep end metavar conclude metavar var ep end metavar <= [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] cut lemma x<=|x| conclude [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | cut lemma leqTransitivity modus ponens metavar var ep end metavar <= [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] modus ponens [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | conclude metavar var ep end metavar <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | cut lemma numericalDifference conclude | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | cut lemma subLeqRight modus ponens | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | = | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | modus ponens metavar var ep end metavar <= | [ metavar var fy end metavar ; metavar var n end metavar ] + - [ metavar var fx end metavar ; metavar var n end metavar ] | conclude metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | cut lemma toNotLess modus ponens metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | conclude not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar cut prop lemma to negated double imply modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens metavar var n end metavar <= metavar var n end metavar modus ponens not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar cut all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed 1rule deduction modus ponens all metavar var m end metavar indeed all metavar var n end metavar indeed all metavar var ep end metavar indeed all metavar var fx end metavar indeed all metavar var fy end metavar indeed for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar infer not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar conclude for all objects metavar var m end metavar indeed not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n end metavar <= metavar var m end metavar imply [ metavar var fx end metavar ; metavar var m end metavar ] <= [ metavar var fy end metavar ; metavar var m end metavar ] + - metavar var ep end metavar imply not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n end metavar <= metavar var n end metavar imply not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | <= metavar var ep end metavar imply not0 not0 | [ metavar var fx end metavar ; metavar var n end metavar ] + - [ metavar var fy end metavar ; metavar var n end metavar ] | = metavar var ep end metavar end quote state proof state cache var c end expand end define