Logiweb aspects of lemma splitNumericalSum in pyk

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The predefined "pyk" aspect

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 end of text end unicode text end text end define

The predefined "tex" aspect

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 end of text end unicode text end text end define

The user defined "the statement aspect" aspect

define statement of lemma splitNumericalSum as system Q infer all metavar var x end metavar indeed all metavar var y 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 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

The user defined "the proof aspect" aspect

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 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar infer lemma splitNumericalSum(++) modus ponens 0 <= metavar var x end metavar modus ponens 0 <= 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 0 <= metavar var x end metavar infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(+-) 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 ) cut all metavar var x end metavar indeed all metavar var y end metavar indeed metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar infer lemma splitNumericalSum(-+) modus ponens metavar var x end metavar <= 0 modus ponens 0 <= 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 metavar var x end metavar <= 0 infer metavar var y end metavar <= 0 infer lemma splitNumericalSum(--) modus ponens metavar var x end metavar <= 0 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 ) 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 0 <= metavar var x end metavar infer 0 <= metavar var y end metavar 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 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar 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 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 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 metavar var x end metavar <= 0 infer 0 <= metavar var y end metavar 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 metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar 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 metavar var x end metavar <= 0 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 metavar var x end metavar <= 0 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 lemma from leqGeq modus ponens 0 <= metavar var x end metavar imply 0 <= metavar var y end metavar 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 metavar var x end metavar <= 0 imply 0 <= metavar var y end metavar 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 y end metavar 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 lemma from leqGeq 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 metavar var x end metavar <= 0 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 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 lemma from leqGeq modus ponens 0 <= metavar var y end metavar 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 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 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