author | wenzelm |
Thu, 26 Jul 2012 17:16:02 +0200 | |
changeset 48519 | 5deda0549f97 |
parent 47183 | doc-src/TutorialI/Types/document/Numbers.tex@f760e15343bc |
child 48611 | b34ff75c23a7 |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
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\def\isabellecontext{Numbers}% |
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% |
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\isadelimtheory |
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% |
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\endisadelimtheory |
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% |
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\isatagtheory |
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\isacommand{theory}\isamarkupfalse% |
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\ Numbers\isanewline |
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\isakeyword{imports}\ Complex{\isaliteral{5F}{\isacharunderscore}}Main\isanewline |
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\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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% |
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\isadelimtheory |
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\isanewline |
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% |
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\endisadelimtheory |
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% |
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\isadelimML |
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\isanewline |
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% |
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\endisadelimML |
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% |
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\isatagML |
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\isacommand{ML}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}Pretty{\isaliteral{2E}{\isachardot}}margin{\isaliteral{5F}{\isacharunderscore}}default\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ {\isadigit{6}}{\isadigit{4}}{\isaliteral{22}{\isachardoublequoteclose}}% |
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\endisatagML |
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{\isafoldML}% |
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% |
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\isadelimML |
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\isanewline |
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% |
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\endisadelimML |
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\isacommand{declare}\isamarkupfalse% |
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\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5B}{\isacharbrackleft}}thy{\isaliteral{5F}{\isacharunderscore}}output{\isaliteral{5F}{\isacharunderscore}}indent\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{5D}{\isacharbrackright}}% |
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\begin{isamarkuptext}% |
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numeric literals; default simprules; can re-orient% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ m\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{2B}{\isacharplus}}\ m{\isaliteral{22}{\isachardoublequoteclose}}% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\isatagproof |
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% |
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\begin{isamarkuptxt}% |
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\begin{isabelle}% |
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ m\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{2B}{\isacharplus}}\ m% |
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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{oops}\isamarkupfalse% |
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% |
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\endisatagproof |
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{\isafoldproof}% |
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% |
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\isadelimproof |
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% |
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\endisadelimproof |
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\isanewline |
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\isanewline |
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\isacommand{fun}\isamarkupfalse% |
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\ h\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{where}\isanewline |
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{\isaliteral{22}{\isachardoublequoteopen}}h\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
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\begin{isamarkuptext}% |
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\isa{h\ {\isadigit{3}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}} |
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\isa{h\ i\ {\isaliteral{3D}{\isacharequal}}\ i}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
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\begin{isabelle}% |
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Numeral{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{numeral_1_eq_1} |
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\begin{isabelle}% |
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{\isadigit{2}}\ {\isaliteral{2B}{\isacharplus}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_2_eq_Suc} |
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\begin{isabelle}% |
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n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}Suc\ n{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_2_eq_Suc'} |
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||
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\begin{isabelle}% |
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a\ {\isaliteral{2B}{\isacharplus}}\ b\ {\isaliteral{2B}{\isacharplus}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_assoc} |
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\begin{isabelle}% |
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a\ {\isaliteral{2B}{\isacharplus}}\ b\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2B}{\isacharplus}}\ a% |
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\end{isabelle} |
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\rulename{add_commute} |
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\begin{isabelle}% |
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b\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2B}{\isacharplus}}\ c{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{add_left_commute} |
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|
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these form add_ac; similarly there is mult_ac% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}Suc{\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ j{\isaliteral{2A}{\isacharasterisk}}l{\isaliteral{2A}{\isacharasterisk}}k\ {\isaliteral{2B}{\isacharplus}}\ m{\isaliteral{2A}{\isacharasterisk}}n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}n{\isaliteral{2A}{\isacharasterisk}}m\ {\isaliteral{2B}{\isacharplus}}\ i\ {\isaliteral{2B}{\isacharplus}}\ k{\isaliteral{2A}{\isacharasterisk}}j{\isaliteral{2A}{\isacharasterisk}}l{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\isatagproof |
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% |
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\begin{isamarkuptxt}% |
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\begin{isabelle}% |
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ l\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{2B}{\isacharplus}}\ m\ {\isaliteral{2A}{\isacharasterisk}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2A}{\isacharasterisk}}\ m\ {\isaliteral{2B}{\isacharplus}}\ i\ {\isaliteral{2B}{\isacharplus}}\ k\ {\isaliteral{2A}{\isacharasterisk}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ l{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ add{\isaliteral{5F}{\isacharunderscore}}ac\ mult{\isaliteral{5F}{\isacharunderscore}}ac{\isaliteral{29}{\isacharparenright}}% |
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\begin{isamarkuptxt}% |
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\begin{isabelle}% |
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ Suc\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2B}{\isacharplus}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2A}{\isacharasterisk}}\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline |
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\isaindent{\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ }f\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2B}{\isacharplus}}\ j\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2A}{\isacharasterisk}}\ l{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{oops}\isamarkupfalse% |
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% |
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\endisatagproof |
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{\isafoldproof}% |
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% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\begin{isamarkuptext}% |
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\begin{isabelle}% |
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m\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ div\ k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ n\ div\ k% |
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\end{isabelle} |
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\rulename{div_le_mono} |
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|
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\begin{isabelle}% |
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{\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2D}{\isacharminus}}\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{3D}{\isacharequal}}\ m\ {\isaliteral{2A}{\isacharasterisk}}\ k\ {\isaliteral{2D}{\isacharminus}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ k% |
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\end{isabelle} |
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\rulename{diff_mult_distrib} |
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|
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\begin{isabelle}% |
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a\ mod\ b\ {\isaliteral{2A}{\isacharasterisk}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ c\ mod\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{mult_mod_left} |
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|
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\begin{isabelle}% |
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P\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2D}{\isacharminus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{3C}{\isacharless}}\ b\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}d{\isaliteral{2E}{\isachardot}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2B}{\isacharplus}}\ d\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ d{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
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\end{isabelle} |
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\rulename{nat_diff_split}% |
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\end{isamarkuptext}% |
|
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{1}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
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% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\isatagproof |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}clarsimp\ split{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{5F}{\isacharunderscore}}diff{\isaliteral{5F}{\isacharunderscore}}split\ iff\ del{\isaliteral{3A}{\isacharcolon}}\ less{\isaliteral{5F}{\isacharunderscore}}Suc{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline |
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\ % |
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\isamarkupcmt{\begin{isabelle}% |
|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}d{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{3C}{\isacharless}}\ Suc\ {\isadigit{0}}{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{3D}{\isacharequal}}\ Suc\ d{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ d\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}% |
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\end{isabelle}% |
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} |
|
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\isanewline |
|
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}subgoal{\isaliteral{5F}{\isacharunderscore}}tac\ {\isaliteral{22}{\isachardoublequoteopen}}n{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ force{\isaliteral{2C}{\isacharcomma}}\ arith{\isaliteral{29}{\isacharparenright}}\isanewline |
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\isacommand{done}\isamarkupfalse% |
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% |
|
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\endisatagproof |
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{\isafoldproof}% |
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% |
|
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\isadelimproof |
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\isanewline |
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% |
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\endisadelimproof |
|
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\isanewline |
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\isanewline |
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\isacommand{lemma}\isamarkupfalse% |
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\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}n\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{4}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 195 |
% |
196 |
\isadelimproof |
|
197 |
% |
|
198 |
\endisadelimproof |
|
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% |
|
200 |
\isatagproof |
|
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\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}simp\ split{\isaliteral{3A}{\isacharcolon}}\ nat{\isaliteral{5F}{\isacharunderscore}}diff{\isaliteral{5F}{\isacharunderscore}}split{\isaliteral{2C}{\isacharcomma}}\ clarify{\isaliteral{29}{\isacharparenright}}\isanewline |
16353 | 203 |
\ % |
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\isamarkupcmt{\begin{isabelle}% |
|
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\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}d{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}n\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{2}}{\isaliteral{3B}{\isacharsemicolon}}\ n\ {\isaliteral{2A}{\isacharasterisk}}\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{4}}\ {\isaliteral{2B}{\isacharplus}}\ d{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ d\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{0}}% |
16353 | 206 |
\end{isabelle}% |
207 |
} |
|
208 |
\isanewline |
|
17175 | 209 |
\isacommand{apply}\isamarkupfalse% |
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\ {\isaliteral{28}{\isacharparenleft}}subgoal{\isaliteral{5F}{\isacharunderscore}}tac\ {\isaliteral{22}{\isachardoublequoteopen}}n{\isaliteral{3D}{\isacharequal}}{\isadigit{0}}\ {\isaliteral{7C}{\isacharbar}}\ n{\isaliteral{3D}{\isacharequal}}{\isadigit{1}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{2C}{\isacharcomma}}\ force{\isaliteral{2C}{\isacharcomma}}\ arith{\isaliteral{29}{\isacharparenright}}\isanewline |
17175 | 211 |
\isacommand{done}\isamarkupfalse% |
212 |
% |
|
17056 | 213 |
\endisatagproof |
214 |
{\isafoldproof}% |
|
215 |
% |
|
216 |
\isadelimproof |
|
217 |
% |
|
218 |
\endisadelimproof |
|
11866 | 219 |
% |
11187 | 220 |
\begin{isamarkuptext}% |
221 |
\begin{isabelle}% |
|
40406 | 222 |
m\ mod\ n\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ m\ {\isaliteral{3C}{\isacharless}}\ n\ then\ m\ else\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{2D}{\isacharminus}}\ n{\isaliteral{29}{\isacharparenright}}\ mod\ n{\isaliteral{29}{\isacharparenright}}% |
11187 | 223 |
\end{isabelle} |
11174 | 224 |
\rulename{mod_if} |
11187 | 225 |
|
226 |
\begin{isabelle}% |
|
40406 | 227 |
a\ div\ b\ {\isaliteral{2A}{\isacharasterisk}}\ b\ {\isaliteral{2B}{\isacharplus}}\ a\ mod\ b\ {\isaliteral{3D}{\isacharequal}}\ a% |
10602 | 228 |
\end{isabelle} |
11174 | 229 |
\rulename{mod_div_equality} |
10602 | 230 |
|
231 |
||
11187 | 232 |
\begin{isabelle}% |
40406 | 233 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ div\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ div\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ div\ c% |
11187 | 234 |
\end{isabelle} |
11174 | 235 |
\rulename{div_mult1_eq} |
10602 | 236 |
|
11187 | 237 |
\begin{isabelle}% |
40406 | 238 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ mod\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ mod\ c% |
11187 | 239 |
\end{isabelle} |
30224 | 240 |
\rulename{mod_mult_right_eq} |
10602 | 241 |
|
11187 | 242 |
\begin{isabelle}% |
40406 | 243 |
a\ div\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ b\ div\ c% |
11187 | 244 |
\end{isabelle} |
11174 | 245 |
\rulename{div_mult2_eq} |
10602 | 246 |
|
11187 | 247 |
\begin{isabelle}% |
40406 | 248 |
a\ mod\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}a\ div\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ mod\ b% |
11187 | 249 |
\end{isabelle} |
11174 | 250 |
\rulename{mod_mult2_eq} |
10602 | 251 |
|
11187 | 252 |
\begin{isabelle}% |
40406 | 253 |
c\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ c\ {\isaliteral{2A}{\isacharasterisk}}\ a\ div\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2A}{\isacharasterisk}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ b% |
11187 | 254 |
\end{isabelle} |
11174 | 255 |
\rulename{div_mult_mult1} |
10602 | 256 |
|
11187 | 257 |
\begin{isabelle}% |
40406 | 258 |
a\ div\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}% |
11187 | 259 |
\end{isabelle} |
27658 | 260 |
\rulename{div_by_0} |
10602 | 261 |
|
11187 | 262 |
\begin{isabelle}% |
40406 | 263 |
a\ mod\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a% |
11187 | 264 |
\end{isabelle} |
27658 | 265 |
\rulename{mod_by_0} |
10602 | 266 |
|
11187 | 267 |
\begin{isabelle}% |
40406 | 268 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}m\ dvd\ n{\isaliteral{3B}{\isacharsemicolon}}\ n\ dvd\ m{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ m\ {\isaliteral{3D}{\isacharequal}}\ n% |
11187 | 269 |
\end{isabelle} |
33750 | 270 |
\rulename{dvd_antisym} |
10602 | 271 |
|
11187 | 272 |
\begin{isabelle}% |
40406 | 273 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}a\ dvd\ b{\isaliteral{3B}{\isacharsemicolon}}\ a\ dvd\ c{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ dvd\ b\ {\isaliteral{2B}{\isacharplus}}\ c% |
11187 | 274 |
\end{isabelle} |
11174 | 275 |
\rulename{dvd_add} |
10602 | 276 |
|
277 |
For the integers, I'd list a few theorems that somehow involve negative |
|
13758 | 278 |
numbers.% |
279 |
\end{isamarkuptext}% |
|
280 |
\isamarkuptrue% |
|
281 |
% |
|
282 |
\begin{isamarkuptext}% |
|
10602 | 283 |
Division, remainder of negatives |
284 |
||
285 |
||
11187 | 286 |
\begin{isabelle}% |
40406 | 287 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ b\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isadigit{0}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ a\ mod\ b% |
11187 | 288 |
\end{isabelle} |
11174 | 289 |
\rulename{pos_mod_sign} |
10602 | 290 |
|
11187 | 291 |
\begin{isabelle}% |
40406 | 292 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ b\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ mod\ b\ {\isaliteral{3C}{\isacharless}}\ b% |
11174 | 293 |
\end{isabelle} |
11187 | 294 |
\rulename{pos_mod_bound} |
10602 | 295 |
|
11187 | 296 |
\begin{isabelle}% |
40406 | 297 |
b\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ mod\ b\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isadigit{0}}% |
11174 | 298 |
\end{isabelle} |
11187 | 299 |
\rulename{neg_mod_sign} |
10602 | 300 |
|
11187 | 301 |
\begin{isabelle}% |
40406 | 302 |
b\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ b\ {\isaliteral{3C}{\isacharless}}\ a\ mod\ b% |
11187 | 303 |
\end{isabelle} |
11174 | 304 |
\rulename{neg_mod_bound} |
10602 | 305 |
|
11187 | 306 |
\begin{isabelle}% |
40406 | 307 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ div\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ div\ c\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}a\ mod\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ div\ c% |
11174 | 308 |
\end{isabelle} |
11187 | 309 |
\rulename{zdiv_zadd1_eq} |
10602 | 310 |
|
11187 | 311 |
\begin{isabelle}% |
40406 | 312 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ mod\ c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ mod\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ mod\ c% |
11187 | 313 |
\end{isabelle} |
30224 | 314 |
\rulename{mod_add_eq} |
10602 | 315 |
|
11187 | 316 |
\begin{isabelle}% |
40406 | 317 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ div\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ div\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ div\ c% |
10602 | 318 |
\end{isabelle} |
11187 | 319 |
\rulename{zdiv_zmult1_eq} |
10602 | 320 |
|
11187 | 321 |
\begin{isabelle}% |
40406 | 322 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ mod\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ mod\ c% |
11187 | 323 |
\end{isabelle} |
47183 | 324 |
\rulename{mod_mult_right_eq} |
10602 | 325 |
|
11187 | 326 |
\begin{isabelle}% |
40406 | 327 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ div\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ div\ b\ div\ c% |
11174 | 328 |
\end{isabelle} |
11187 | 329 |
\rulename{zdiv_zmult2_eq} |
10602 | 330 |
|
11187 | 331 |
\begin{isabelle}% |
40406 | 332 |
{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ c\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ a\ mod\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}a\ div\ b\ mod\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2B}{\isacharplus}}\ a\ mod\ b% |
11187 | 333 |
\end{isabelle} |
14400 | 334 |
\rulename{zmod_zmult2_eq}% |
11187 | 335 |
\end{isamarkuptext}% |
17175 | 336 |
\isamarkuptrue% |
337 |
\isacommand{lemma}\isamarkupfalse% |
|
40406 | 338 |
\ {\isaliteral{22}{\isachardoublequoteopen}}abs\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2B}{\isacharplus}}y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ abs\ x\ {\isaliteral{2B}{\isacharplus}}\ abs\ {\isaliteral{28}{\isacharparenleft}}y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 339 |
% |
340 |
\isadelimproof |
|
341 |
% |
|
342 |
\endisadelimproof |
|
343 |
% |
|
344 |
\isatagproof |
|
17175 | 345 |
\isacommand{by}\isamarkupfalse% |
346 |
\ arith% |
|
17056 | 347 |
\endisatagproof |
348 |
{\isafoldproof}% |
|
349 |
% |
|
350 |
\isadelimproof |
|
351 |
\isanewline |
|
352 |
% |
|
353 |
\endisadelimproof |
|
15481 | 354 |
\isanewline |
17175 | 355 |
\isacommand{lemma}\isamarkupfalse% |
40406 | 356 |
\ {\isaliteral{22}{\isachardoublequoteopen}}abs\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}{\isaliteral{2A}{\isacharasterisk}}x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}\ {\isaliteral{2A}{\isacharasterisk}}\ abs\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 357 |
% |
358 |
\isadelimproof |
|
359 |
% |
|
360 |
\endisadelimproof |
|
361 |
% |
|
362 |
\isatagproof |
|
17175 | 363 |
\isacommand{by}\isamarkupfalse% |
40406 | 364 |
\ {\isaliteral{28}{\isacharparenleft}}simp\ add{\isaliteral{3A}{\isacharcolon}}\ abs{\isaliteral{5F}{\isacharunderscore}}if{\isaliteral{29}{\isacharparenright}}% |
17056 | 365 |
\endisatagproof |
366 |
{\isafoldproof}% |
|
367 |
% |
|
368 |
\isadelimproof |
|
369 |
% |
|
370 |
\endisadelimproof |
|
11866 | 371 |
% |
11187 | 372 |
\begin{isamarkuptext}% |
13758 | 373 |
Induction rules for the Integers |
374 |
||
375 |
\begin{isabelle}% |
|
40406 | 376 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ k{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{5C3C6C653E}{\isasymle}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 377 |
\end{isabelle} |
378 |
\rulename{int_ge_induct} |
|
379 |
||
380 |
\begin{isabelle}% |
|
40406 | 381 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{3C}{\isacharless}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}k\ {\isaliteral{3C}{\isacharless}}\ i{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 382 |
\end{isabelle} |
383 |
\rulename{int_gr_induct} |
|
384 |
||
385 |
\begin{isabelle}% |
|
40406 | 386 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{5C3C6C653E}{\isasymle}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ k{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{5C3C6C653E}{\isasymle}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 387 |
\end{isabelle} |
388 |
\rulename{int_le_induct} |
|
389 |
||
390 |
\begin{isabelle}% |
|
40406 | 391 |
{\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{3C}{\isacharless}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{3B}{\isacharsemicolon}}\ {\isaliteral{5C3C416E643E}{\isasymAnd}}i{\isaliteral{2E}{\isachardot}}\ {\isaliteral{5C3C6C6272616B6B3E}{\isasymlbrakk}}i\ {\isaliteral{3C}{\isacharless}}\ k{\isaliteral{3B}{\isacharsemicolon}}\ P\ i{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C726272616B6B3E}{\isasymrbrakk}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ P\ i% |
13758 | 392 |
\end{isabelle} |
393 |
\rulename{int_less_induct}% |
|
394 |
\end{isamarkuptext}% |
|
395 |
\isamarkuptrue% |
|
396 |
% |
|
397 |
\begin{isamarkuptext}% |
|
14400 | 398 |
FIELDS |
10776 | 399 |
|
11187 | 400 |
\begin{isabelle}% |
40406 | 401 |
x\ {\isaliteral{3C}{\isacharless}}\ y\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C6578697374733E}{\isasymexists}}z{\isaliteral{3E}{\isachargreater}}x{\isaliteral{2E}{\isachardot}}\ z\ {\isaliteral{3C}{\isacharless}}\ y% |
11187 | 402 |
\end{isabelle} |
14295 | 403 |
\rulename{dense} |
10776 | 404 |
|
11187 | 405 |
\begin{isabelle}% |
40406 | 406 |
a\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2F}{\isacharslash}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ {\isaliteral{2F}{\isacharslash}}\ c% |
11187 | 407 |
\end{isabelle} |
14288 | 408 |
\rulename{times_divide_eq_right} |
10776 | 409 |
|
11187 | 410 |
\begin{isabelle}% |
40406 | 411 |
b\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{2A}{\isacharasterisk}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ a\ {\isaliteral{2F}{\isacharslash}}\ c% |
11187 | 412 |
\end{isabelle} |
14288 | 413 |
\rulename{times_divide_eq_left} |
10776 | 414 |
|
11187 | 415 |
\begin{isabelle}% |
40406 | 416 |
a\ {\isaliteral{2F}{\isacharslash}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2F}{\isacharslash}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2A}{\isacharasterisk}}\ c\ {\isaliteral{2F}{\isacharslash}}\ b% |
11187 | 417 |
\end{isabelle} |
14288 | 418 |
\rulename{divide_divide_eq_right} |
10776 | 419 |
|
11187 | 420 |
\begin{isabelle}% |
40406 | 421 |
a\ {\isaliteral{2F}{\isacharslash}}\ b\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2F}{\isacharslash}}\ {\isaliteral{28}{\isacharparenleft}}b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}% |
11187 | 422 |
\end{isabelle} |
14288 | 423 |
\rulename{divide_divide_eq_left} |
10776 | 424 |
|
11187 | 425 |
\begin{isabelle}% |
40406 | 426 |
{\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2F}{\isacharslash}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{2D}{\isacharminus}}\ a\ {\isaliteral{2F}{\isacharslash}}\ b% |
11187 | 427 |
\end{isabelle} |
14295 | 428 |
\rulename{minus_divide_left} |
10776 | 429 |
|
11187 | 430 |
\begin{isabelle}% |
40406 | 431 |
{\isaliteral{2D}{\isacharminus}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2F}{\isacharslash}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2F}{\isacharslash}}\ {\isaliteral{2D}{\isacharminus}}\ b% |
11187 | 432 |
\end{isabelle} |
14295 | 433 |
\rulename{minus_divide_right} |
10776 | 434 |
|
435 |
This last NOT a simprule |
|
436 |
||
11187 | 437 |
\begin{isabelle}% |
40406 | 438 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{2F}{\isacharslash}}\ c\ {\isaliteral{2B}{\isacharplus}}\ b\ {\isaliteral{2F}{\isacharslash}}\ c% |
11174 | 439 |
\end{isabelle} |
14295 | 440 |
\rulename{add_divide_distrib}% |
11187 | 441 |
\end{isamarkuptext}% |
17175 | 442 |
\isamarkuptrue% |
443 |
\isacommand{lemma}\isamarkupfalse% |
|
40406 | 444 |
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isadigit{3}}{\isaliteral{2F}{\isacharslash}}{\isadigit{4}}\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{7}}{\isaliteral{2F}{\isacharslash}}{\isadigit{8}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ real{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline |
17056 | 445 |
% |
446 |
\isadelimproof |
|
447 |
% |
|
448 |
\endisadelimproof |
|
449 |
% |
|
450 |
\isatagproof |
|
17175 | 451 |
\isacommand{by}\isamarkupfalse% |
452 |
\ simp% |
|
17056 | 453 |
\endisatagproof |
454 |
{\isafoldproof}% |
|
455 |
% |
|
456 |
\isadelimproof |
|
457 |
\ \isanewline |
|
458 |
% |
|
459 |
\endisadelimproof |
|
11174 | 460 |
\isanewline |
17175 | 461 |
\isacommand{lemma}\isamarkupfalse% |
40406 | 462 |
\ {\isaliteral{22}{\isachardoublequoteopen}}P\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{3}}{\isaliteral{2F}{\isacharslash}}{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}{\isaliteral{2F}{\isacharslash}}{\isadigit{1}}{\isadigit{5}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ real{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
17056 | 463 |
\isadelimproof |
464 |
% |
|
465 |
\endisadelimproof |
|
466 |
% |
|
467 |
\isatagproof |
|
16353 | 468 |
% |
469 |
\begin{isamarkuptxt}% |
|
470 |
\begin{isabelle}% |
|
40406 | 471 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ P\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{3}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{4}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
16353 | 472 |
\end{isabelle}% |
473 |
\end{isamarkuptxt}% |
|
17175 | 474 |
\isamarkuptrue% |
475 |
\isacommand{apply}\isamarkupfalse% |
|
476 |
\ simp% |
|
16353 | 477 |
\begin{isamarkuptxt}% |
478 |
\begin{isabelle}% |
|
40406 | 479 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ P\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{2}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{5}}{\isaliteral{29}{\isacharparenright}}% |
16353 | 480 |
\end{isabelle}% |
481 |
\end{isamarkuptxt}% |
|
17175 | 482 |
\isamarkuptrue% |
483 |
\isacommand{oops}\isamarkupfalse% |
|
484 |
% |
|
17056 | 485 |
\endisatagproof |
486 |
{\isafoldproof}% |
|
487 |
% |
|
488 |
\isadelimproof |
|
489 |
% |
|
490 |
\endisadelimproof |
|
491 |
\isanewline |
|
15481 | 492 |
\isanewline |
17175 | 493 |
\isacommand{lemma}\isamarkupfalse% |
40406 | 494 |
\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{3}}{\isaliteral{2F}{\isacharslash}}{\isadigit{4}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}{\isaliteral{2F}{\isacharslash}}{\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ {\isaliteral{28}{\isacharparenleft}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ real{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}% |
17056 | 495 |
\isadelimproof |
496 |
% |
|
497 |
\endisadelimproof |
|
498 |
% |
|
499 |
\isatagproof |
|
16353 | 500 |
% |
501 |
\begin{isamarkuptxt}% |
|
502 |
\begin{isabelle}% |
|
40406 | 503 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{3}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{4}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{8}}\ {\isaliteral{2F}{\isacharslash}}\ {\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ x% |
16353 | 504 |
\end{isabelle}% |
505 |
\end{isamarkuptxt}% |
|
17175 | 506 |
\isamarkuptrue% |
507 |
\isacommand{apply}\isamarkupfalse% |
|
508 |
\ simp% |
|
16353 | 509 |
\begin{isamarkuptxt}% |
510 |
\begin{isabelle}% |
|
40406 | 511 |
\ {\isadigit{1}}{\isaliteral{2E}{\isachardot}}\ {\isadigit{2}}\ {\isaliteral{3C}{\isacharless}}\ x\ {\isaliteral{2A}{\isacharasterisk}}\ {\isadigit{5}}% |
16353 | 512 |
\end{isabelle}% |
513 |
\end{isamarkuptxt}% |
|
17175 | 514 |
\isamarkuptrue% |
515 |
\isacommand{oops}\isamarkupfalse% |
|
516 |
% |
|
17056 | 517 |
\endisatagproof |
518 |
{\isafoldproof}% |
|
519 |
% |
|
520 |
\isadelimproof |
|
521 |
% |
|
522 |
\endisadelimproof |
|
14400 | 523 |
% |
524 |
\begin{isamarkuptext}% |
|
525 |
Ring and Field |
|
526 |
||
527 |
Requires a field, or else an ordered ring |
|
528 |
||
529 |
\begin{isabelle}% |
|
40406 | 530 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2A}{\isacharasterisk}}\ b\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ b\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}% |
14400 | 531 |
\end{isabelle} |
532 |
\rulename{mult_eq_0_iff} |
|
533 |
||
534 |
\begin{isabelle}% |
|
40406 | 535 |
{\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{2A}{\isacharasterisk}}\ c\ {\isaliteral{3D}{\isacharequal}}\ b\ {\isaliteral{2A}{\isacharasterisk}}\ c{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}% |
14400 | 536 |
\end{isabelle} |
537 |
\rulename{mult_cancel_right} |
|
538 |
||
539 |
\begin{isabelle}% |
|
40406 | 540 |
{\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{2A}{\isacharasterisk}}\ a\ {\isaliteral{3D}{\isacharequal}}\ c\ {\isaliteral{2A}{\isacharasterisk}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}% |
14400 | 541 |
\end{isabelle} |
23504 | 542 |
\rulename{mult_cancel_left}% |
14400 | 543 |
\end{isamarkuptext}% |
17175 | 544 |
\isamarkuptrue% |
17056 | 545 |
% |
14400 | 546 |
\begin{isamarkuptext}% |
547 |
effect of show sorts on the above |
|
548 |
||
549 |
\begin{isabelle}% |
|
40406 | 550 |
{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}c{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline |
551 |
\isaindent{{\isaliteral{28}{\isacharparenleft}}}c\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{28}{\isacharparenleft}}b{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline |
|
552 |
{\isaliteral{28}{\isacharparenleft}}c\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{5C3C436F6C6F6E3E}{\isasymColon}}ring{\isaliteral{5F}{\isacharunderscore}}no{\isaliteral{5F}{\isacharunderscore}}zero{\isaliteral{5F}{\isacharunderscore}}divisors{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ a\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{29}{\isacharparenright}}% |
|
14400 | 553 |
\end{isabelle} |
23504 | 554 |
\rulename{mult_cancel_left}% |
14400 | 555 |
\end{isamarkuptext}% |
17175 | 556 |
\isamarkuptrue% |
17056 | 557 |
% |
14400 | 558 |
\begin{isamarkuptext}% |
559 |
absolute value |
|
560 |
||
561 |
\begin{isabelle}% |
|
40406 | 562 |
{\isaliteral{5C3C6261723E}{\isasymbar}}a\ {\isaliteral{2A}{\isacharasterisk}}\ b{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{2A}{\isacharasterisk}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}b{\isaliteral{5C3C6261723E}{\isasymbar}}% |
14400 | 563 |
\end{isabelle} |
564 |
\rulename{abs_mult} |
|
565 |
||
566 |
\begin{isabelle}% |
|
40406 | 567 |
{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}a\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b\ {\isaliteral{5C3C616E643E}{\isasymand}}\ {\isaliteral{2D}{\isacharminus}}\ a\ {\isaliteral{5C3C6C653E}{\isasymle}}\ b{\isaliteral{29}{\isacharparenright}}% |
14400 | 568 |
\end{isabelle} |
569 |
\rulename{abs_le_iff} |
|
570 |
||
571 |
\begin{isabelle}% |
|
40406 | 572 |
{\isaliteral{5C3C6261723E}{\isasymbar}}a\ {\isaliteral{2B}{\isacharplus}}\ b{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}b{\isaliteral{5C3C6261723E}{\isasymbar}}% |
14400 | 573 |
\end{isabelle} |
574 |
\rulename{abs_triangle_ineq} |
|
575 |
||
576 |
\begin{isabelle}% |
|
40406 | 577 |
a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\ {\isaliteral{2B}{\isacharplus}}\ n\isaliteral{5C3C5E657375703E}{}\isactrlesup \ {\isaliteral{3D}{\isacharequal}}\ a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\isaliteral{5C3C5E657375703E}{}\isactrlesup \ {\isaliteral{2A}{\isacharasterisk}}\ a\isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup % |
14400 | 578 |
\end{isabelle} |
579 |
\rulename{power_add} |
|
580 |
||
581 |
\begin{isabelle}% |
|
40406 | 582 |
a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\ {\isaliteral{2A}{\isacharasterisk}}\ n\isaliteral{5C3C5E657375703E}{}\isactrlesup \ {\isaliteral{3D}{\isacharequal}}\ a\isaliteral{5C3C5E627375703E}{}\isactrlbsup m\isaliteral{5C3C5E657375703E}{}\isactrlesup \isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup % |
14400 | 583 |
\end{isabelle} |
584 |
\rulename{power_mult} |
|
585 |
||
586 |
\begin{isabelle}% |
|
40406 | 587 |
{\isaliteral{5C3C6261723E}{\isasymbar}}a\isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup {\isaliteral{5C3C6261723E}{\isasymbar}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C6261723E}{\isasymbar}}a{\isaliteral{5C3C6261723E}{\isasymbar}}\isaliteral{5C3C5E627375703E}{}\isactrlbsup n\isaliteral{5C3C5E657375703E}{}\isactrlesup % |
14400 | 588 |
\end{isabelle} |
589 |
\rulename{power_abs}% |
|
590 |
\end{isamarkuptext}% |
|
17175 | 591 |
\isamarkuptrue% |
17056 | 592 |
% |
593 |
\isadelimtheory |
|
594 |
% |
|
595 |
\endisadelimtheory |
|
596 |
% |
|
597 |
\isatagtheory |
|
17175 | 598 |
\isacommand{end}\isamarkupfalse% |
599 |
% |
|
17056 | 600 |
\endisatagtheory |
601 |
{\isafoldtheory}% |
|
602 |
% |
|
603 |
\isadelimtheory |
|
604 |
% |
|
605 |
\endisadelimtheory |
|
606 |
\isanewline |
|
11187 | 607 |
\end{isabellebody}% |
10602 | 608 |
%%% Local Variables: |
609 |
%%% mode: latex |
|
610 |
%%% TeX-master: "root" |
|
611 |
%%% End: |