Logiweb aspects of lemma preserveLessGreater in pyk

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The predefined "pyk" aspect

define pyk of lemma preserveLessGreater 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 p unicode small r unicode small e unicode small s unicode small e unicode small r unicode small v unicode small e unicode capital l unicode small e unicode small s unicode small s unicode capital g unicode small r unicode small e unicode small a unicode small t 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 preserveLessGreater as text unicode start of text unicode capital p unicode small r unicode small e unicode small s unicode small e unicode small r unicode small v unicode small e unicode capital l unicode small e unicode small s unicode small s unicode capital g unicode small r unicode small e unicode small a unicode small t unicode small e unicode small r unicode end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma preserveLessGreater as system Q infer all metavar var x1 end metavar indeed all metavar var x2 end metavar indeed all metavar var y1 end metavar indeed all metavar var y2 end metavar indeed all metavar var z end metavar indeed metavar var x1 end metavar <= metavar var y1 end metavar + - metavar var z end metavar infer not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end define

The user defined "the proof aspect" aspect

define proof of lemma preserveLessGreater as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x1 end metavar indeed all metavar var x2 end metavar indeed all metavar var y1 end metavar indeed all metavar var y2 end metavar indeed all metavar var z end metavar indeed metavar var x1 end metavar <= metavar var y1 end metavar + - metavar var z end metavar infer not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar infer lemma 0<=|x| conclude 0 <= | metavar var x1 end metavar + - metavar var x2 end metavar | cut lemma leqLessTransitivity modus ponens 0 <= | metavar var x1 end metavar + - metavar var x2 end metavar | modus ponens not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma positiveTripled modus ponens not0 0 <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 0 = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 0 <= metavar var z end metavar imply not0 not0 0 = metavar var z end metavar cut lemma numericalDifferenceLess modus ponens not0 | metavar var x1 end metavar + - metavar var x2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var x1 end metavar + - metavar var x2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar imply not0 not0 metavar var x1 end metavar <= metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x1 end metavar = metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut prop lemma first conjunct modus ponens not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar imply not0 not0 metavar var x1 end metavar <= metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x1 end metavar = metavar var x2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar cut lemma negativeToRight(Less) modus ponens not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var x1 end metavar imply not0 not0 metavar var x2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var x1 end metavar conclude not0 metavar var x2 end metavar <= metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma leqAddition modus ponens metavar var x1 end metavar <= metavar var y1 end metavar + - metavar var z end metavar conclude metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma -x+(1/3)x=-(2/3)x conclude - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma three2twoTerms modus ponens - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma subLeqRight modus ponens metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - metavar var z end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma numericalDifferenceLess modus ponens not0 | metavar var y1 end metavar + - metavar var y2 end metavar | <= 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 | metavar var y1 end metavar + - metavar var y2 end metavar | = 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar imply not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar imply not0 not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut prop lemma second conjunct modus ponens not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar imply not0 not0 metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar imply not0 not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma positiveToLeft(Less) modus ponens not0 metavar var y1 end metavar <= metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar = metavar var y2 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar cut lemma lessAddition modus ponens not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar conclude not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut axiom plusAssociativity conclude metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma -(1/3)x-(1/3)x=-(2/3)x conclude - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma eqAdditionLeft modus ponens - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma eqTransitivity modus ponens metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma subLessLeft modus ponens metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma lessLeqTransitivity modus ponens not0 metavar var x2 end metavar <= metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens metavar var x1 end metavar + 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var x2 end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma lessTransitivity modus ponens not0 metavar var x2 end metavar <= metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar modus ponens not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var y1 end metavar + - 1 + 1 * 1/ 1 + 1 + 1 * metavar var z end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar conclude not0 metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar cut lemma lessLeq modus ponens not0 metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar imply not0 not0 metavar var x2 end metavar = metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar conclude metavar var x2 end metavar <= metavar var y2 end metavar + - 1/ 1 + 1 + 1 * metavar var z end metavar end quote state proof state cache var c end expand end define