define pyk of lemma insertMiddleTerm(Numerical) 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 i unicode small n unicode small s unicode small e unicode small r unicode small t unicode capital m unicode small i unicode small d unicode small d unicode small l unicode small e unicode capital t unicode small e unicode small r unicode small m unicode left parenthesis 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 right parenthesis unicode end of text end unicode text end text end define
define tex of lemma insertMiddleTerm(Numerical) as text unicode start of text unicode small i unicode small n unicode small s unicode small e unicode small r unicode small t unicode capital m unicode small i unicode small d unicode small d unicode small l unicode small e unicode capital t unicode small e unicode small r unicode small m unicode left parenthesis 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 right parenthesis unicode end of text end unicode text end text end define
define statement of lemma insertMiddleTerm(Numerical) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) end define
define proof of lemma insertMiddleTerm(Numerical) as lambda var c dot lambda var x dot proof expand quote system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed all metavar var z end metavar indeed lemma splitNumericalSum conclude if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) cut lemma insertMiddleTerm(Sum) conclude metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar cut lemma sameNumerical modus ponens metavar var x end metavar + metavar var y end metavar = metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) cut lemma eqSymmetry modus ponens if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) = if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) modus ponens if( 0 <= metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar , - metavar var x end metavar + - metavar var z end metavar + metavar var z end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar + metavar var y end metavar , metavar var x end metavar + metavar var y end metavar , - metavar var x end metavar + metavar var y end metavar ) <= if( 0 <= metavar var x end metavar + - metavar var z end metavar , metavar var x end metavar + - metavar var z end metavar , - metavar var x end metavar + - metavar var z end metavar ) + if( 0 <= metavar var z end metavar + metavar var y end metavar , metavar var z end metavar + metavar var y end metavar , - metavar var z end metavar + metavar var y end metavar ) end quote state proof state cache var c end expand end define
The pyk compiler, version 0.grue.20060417+ by Klaus Grue,