Logiweb aspects of lemma toLess(F) helper in pyk

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The predefined "pyk" aspect

define pyk of lemma toLess(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 t unicode small o unicode capital l unicode small e unicode small s unicode small s 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 toLess(F) helper as text unicode start of text unicode capital t unicode small o unicode capital l unicode small e unicode small s unicode small s 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 toLess(F) helper as system Q infer all metavar var m end metavar indeed all metavar var n1 end metavar indeed all metavar var n2 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 n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx 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 not0 if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar imply [ metavar var fy end metavar ; metavar var m end metavar ] <= [ metavar var fx end metavar ; metavar var m end metavar ] + - metavar var ep end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma toLess(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 n1 end metavar indeed all metavar var n2 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 n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | infer not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar infer if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar infer lemma leqMax1 conclude metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) cut lemma leqTransitivity modus ponens metavar var n1 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar conclude metavar var n1 end metavar <= metavar var m end metavar cut lemma leqMax2 conclude metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) cut lemma leqTransitivity modus ponens metavar var n2 end metavar <= if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) modus ponens if( metavar var n2 end metavar <= metavar var n1 end metavar , metavar var n1 end metavar , metavar var n2 end metavar ) <= metavar var m end metavar conclude metavar var n2 end metavar <= metavar var m end metavar cut lemma a4 at metavar var m 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 n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply not0 metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m 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 n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m 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 n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut 1rule mp modus ponens metavar var n1 end metavar <= metavar var m end metavar imply not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | modus ponens metavar var n1 end metavar <= metavar var m end metavar conclude not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | cut lemma fromNumericalGreater modus ponens not0 metavar var ep end metavar <= | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | imply not0 not0 metavar var ep end metavar = | [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] | conclude not0 not0 [ 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 [ 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 metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut prop lemma mp2 modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar imply metavar var n2 end metavar <= metavar var m end metavar imply not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar modus ponens not0 0 <= metavar var ep end metavar imply not0 not0 0 = metavar var ep end metavar modus ponens metavar var n2 end metavar <= metavar var m end metavar conclude not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar cut lemma lessNegated modus ponens not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] <= metavar var ep end metavar imply not0 not0 [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = metavar var ep end metavar conclude not0 - metavar var ep end metavar <= - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] cut lemma minusNegated conclude - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma subLessRight modus ponens - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens not0 - metavar var ep end metavar <= - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = - [ metavar var fy end metavar ; metavar var m end metavar ] + - [ metavar var fx end metavar ; metavar var m end metavar ] conclude not0 - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma lessLeq modus ponens not0 - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 - metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] conclude - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma toNotLess modus ponens - metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] conclude not0 not0 [ 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 [ 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 cut prop lemma negate first disjunct modus ponens not0 not0 [ 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 [ 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 metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] modus ponens not0 not0 [ 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 [ 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 metavar var ep end metavar <= [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] imply not0 not0 metavar var ep end metavar = [ metavar var fx end metavar ; metavar var m end metavar ] + - [ metavar var fy end metavar ; metavar var m end metavar ] cut lemma switchTerms(x