Logiweb aspects of lemma nonpositiveNumerical in pyk

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The predefined "pyk" aspect

define pyk of lemma nonpositiveNumerical 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 n unicode small o unicode small n unicode small p unicode small o unicode small s unicode small i unicode small t unicode small i unicode small v unicode small e 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 end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma nonpositiveNumerical as system Q infer all metavar var x end metavar indeed metavar var x end metavar <= 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma nonpositiveNumerical as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer lemma negativeNumerical modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut all metavar var x end metavar indeed metavar var x end metavar = 0 infer lemma eqSymmetry modus ponens metavar var x end metavar = 0 conclude 0 = metavar var x end metavar cut lemma eqLeq modus ponens 0 = metavar var x end metavar conclude 0 <= metavar var x end metavar cut lemma nonnegativeNumerical modus ponens 0 <= metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar cut lemma -0=0 conclude - 0 = 0 cut lemma eqSymmetry modus ponens - 0 = 0 conclude 0 = - 0 cut lemma eqNegated modus ponens 0 = metavar var x end metavar conclude - 0 = - metavar var x end metavar cut lemma eqTransitivity5 modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = metavar var x end metavar modus ponens metavar var x end metavar = 0 modus ponens 0 = - 0 modus ponens - 0 = - metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut all metavar var x end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut 1rule deduction modus ponens all metavar var x end metavar indeed metavar var x end metavar = 0 infer if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar cut metavar var x end metavar <= 0 infer lemma leqLessEq modus ponens metavar var x end metavar <= 0 conclude not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply metavar var x end metavar = 0 cut prop lemma from disjuncts modus ponens not0 not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply metavar var x end metavar = 0 modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar modus ponens metavar var x end metavar = 0 imply if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = - metavar var x end metavar end quote state proof state cache var c end expand end define