Logiweb aspects of lemma 0<=|x| in pyk

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The predefined "pyk" aspect

define pyk of lemma 0<=|x| as text unicode start of text unicode small l unicode small e unicode small m unicode small m unicode small a unicode space unicode zero unicode less than unicode equal sign unicode vertical line unicode small x unicode vertical line unicode end of text end unicode text end text end define

The predefined "tex" aspect

define tex of lemma 0<=|x| as text unicode start of text unicode zero unicode less than unicode equal sign unicode vertical line unicode small x unicode vertical line unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma 0<=|x| as system Q infer all metavar var x end metavar indeed 0 <= if( 0 <= 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 0<=|x| as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed 0 <= metavar var x end metavar infer 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 eqSymmetry modus ponens 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 = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma subLeqRight modus ponens metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens 0 <= metavar var x end metavar conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut all metavar var x end metavar indeed not0 0 <= metavar var x end metavar infer lemma toLess modus ponens not0 0 <= metavar var x end metavar conclude not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 cut 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 lemma eqSymmetry modus ponens 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 = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma negativeNegated modus ponens not0 metavar var x end metavar <= 0 imply not0 not0 metavar var x end metavar = 0 conclude not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar cut lemma lessLeq modus ponens not0 0 <= - metavar var x end metavar imply not0 not0 0 = - metavar var x end metavar conclude 0 <= - metavar var x end metavar cut lemma subLeqRight modus ponens - metavar var x end metavar = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens 0 <= - metavar var x end metavar conclude 0 <= if( 0 <= 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 0 <= metavar var x end metavar infer 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 <= metavar var x end metavar imply 0 <= if( 0 <= 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 not0 0 <= metavar var x end metavar infer 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude not0 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut prop lemma from negations modus ponens 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens not0 0 <= metavar var x end metavar imply 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 <= if( 0 <= 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