define pyk of lemma splitNumericalSum(+-) 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 s unicode small p unicode small l unicode small i unicode small t 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 capital s unicode small u unicode small m unicode left parenthesis unicode plus sign unicode hyphen unicode right parenthesis unicode end of text end unicode text end text end define
define tex of lemma splitNumericalSum(+-) as text unicode start of text unicode small s unicode small p unicode small l unicode small i unicode small t 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 capital s unicode small u unicode small m unicode left parenthesis unicode plus sign unicode hyphen unicode right parenthesis unicode end of text end unicode text end text end define
define statement of lemma splitNumericalSum(+-) as system Q infer all metavar var x end metavar indeed all metavar var y end metavar indeed 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) end define
define proof of lemma splitNumericalSum(+-) 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 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-, smallNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= 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 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 x end metavar , - metavar var x end metavar ) cut lemma 0<=|x| conclude 0 <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma leqAdditionLeft modus ponens 0 <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut axiom plus0 conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma subLeqLeft modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 = if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma leqTransitivity 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 x end metavar , - metavar var x end metavar ) modus ponens if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma toLess modus ponens not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma splitNumericalSum(+-, bigNegative) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 modus ponens not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) <= if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) imply not0 not0 if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) = if( 0 <= metavar var y end metavar , metavar var y 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 y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma 0<=|x| conclude 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) cut lemma leqAddition modus ponens 0 <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) conclude 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma plus0Left conclude 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma subLeqLeft modus ponens 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) = if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens 0 + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut lemma leqTransitivity 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 y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 1rule deduction modus ponens all metavar var x end metavar indeed all metavar var y end metavar indeed not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) infer 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer prop lemma from negations modus ponens if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens not0 if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) <= if( 0 <= metavar var x end metavar , metavar var x end metavar , - metavar var x end metavar ) imply 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) conclude 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) cut prop lemma mp2 modus ponens 0 <= metavar var x end metavar imply metavar var y end metavar <= 0 imply 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y end metavar , - metavar var y end metavar ) modus ponens 0 <= metavar var x end metavar modus ponens metavar var y end metavar <= 0 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 x end metavar , - metavar var x end metavar ) + if( 0 <= metavar var y end metavar , metavar var y 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,