author | paulson |
Wed, 06 Dec 2000 11:47:01 +0100 | |
changeset 10602 | 49109fe3e919 |
child 10617 | adc0ed64a120 |
permissions | -rw-r--r-- |
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% |
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\begin{isabellebody}% |
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\def\isabellecontext{Numbers}% |
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\isanewline |
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\isacommand{theory}\ Numbers\ {\isacharequal}\ Main{\isacharcolon}\isanewline |
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\isanewline |
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\isacommand{ML}\ {\isachardoublequote}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequote}% |
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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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\isacommand{lemma}\ {\isachardoublequote}{\isacharhash}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequote}\isanewline |
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\isacommand{oops}% |
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\begin{isamarkuptext}% |
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proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{0}}\isanewline |
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\isanewline |
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goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline |
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{\isacharparenleft}{\isacharhash}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m\isanewline |
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\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isacharhash}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m |
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|
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}% |
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\end{isabelle} |
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\rulename{numeral_0_eq_0} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{1}}\ {\isacharequal}\ {\isadigit{1}}% |
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\end{isabelle} |
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\rulename{numeral_1_eq_1} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{2}}\ {\isacharplus}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\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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\ \ \ \ \ n\ {\isacharplus}\ {\isacharhash}{\isadigit{2}}\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\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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\ \ \ \ \ m\ {\isacharplus}\ n\ {\isacharplus}\ k\ {\isacharequal}\ m\ {\isacharplus}\ {\isacharparenleft}n\ {\isacharplus}\ k{\isacharparenright}% |
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\end{isabelle} |
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\rulename{add_assoc} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ m\ {\isacharplus}\ n\ {\isacharequal}\ n\ {\isacharplus}\ m% |
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\end{isabelle} |
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\rulename{add_commute} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ x\ {\isacharplus}\ {\isacharparenleft}y\ {\isacharplus}\ z{\isacharparenright}\ {\isacharequal}\ y\ {\isacharplus}\ {\isacharparenleft}x\ {\isacharplus}\ z{\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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\isacommand{lemma}\ {\isachardoublequote}Suc{\isacharparenleft}i\ {\isacharplus}\ j{\isacharasterisk}l{\isacharasterisk}k\ {\isacharplus}\ m{\isacharasterisk}n{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}n{\isacharasterisk}m\ {\isacharplus}\ i\ {\isacharplus}\ k{\isacharasterisk}j{\isacharasterisk}l{\isacharparenright}{\isachardoublequote}\isanewline |
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\isacommand{apply}\ {\isacharparenleft}simp\ add{\isacharcolon}\ add{\isacharunderscore}ac\ mult{\isacharunderscore}ac{\isacharparenright}\isanewline |
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\isacommand{oops}% |
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\begin{isamarkuptext}% |
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proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{0}}\isanewline |
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\isanewline |
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goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline |
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Suc\ {\isacharparenleft}i\ {\isacharplus}\ j\ {\isacharasterisk}\ l\ {\isacharasterisk}\ k\ {\isacharplus}\ m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}n\ {\isacharasterisk}\ m\ {\isacharplus}\ i\ {\isacharplus}\ k\ {\isacharasterisk}\ j\ {\isacharasterisk}\ l{\isacharparenright}\isanewline |
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\ {\isadigit{1}}{\isachardot}\ Suc\ {\isacharparenleft}i\ {\isacharplus}\ j\ {\isacharasterisk}\ l\ {\isacharasterisk}\ k\ {\isacharplus}\ m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}n\ {\isacharasterisk}\ m\ {\isacharplus}\ i\ {\isacharplus}\ k\ {\isacharasterisk}\ j\ {\isacharasterisk}\ l{\isacharparenright} |
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|
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proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline |
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\isanewline |
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goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline |
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Suc\ {\isacharparenleft}i\ {\isacharplus}\ j\ {\isacharasterisk}\ l\ {\isacharasterisk}\ k\ {\isacharplus}\ m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ f\ {\isacharparenleft}n\ {\isacharasterisk}\ m\ {\isacharplus}\ i\ {\isacharplus}\ k\ {\isacharasterisk}\ j\ {\isacharasterisk}\ l{\isacharparenright}\isanewline |
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\ {\isadigit{1}}{\isachardot}\ Suc\ {\isacharparenleft}i\ {\isacharplus}\ {\isacharparenleft}m\ {\isacharasterisk}\ n\ {\isacharplus}\ j\ {\isacharasterisk}\ {\isacharparenleft}k\ {\isacharasterisk}\ l{\isacharparenright}{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline |
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\ \ \ \ f\ {\isacharparenleft}i\ {\isacharplus}\ {\isacharparenleft}m\ {\isacharasterisk}\ n\ {\isacharplus}\ j\ {\isacharasterisk}\ {\isacharparenleft}k\ {\isacharasterisk}\ l{\isacharparenright}{\isacharparenright}{\isacharparenright}% |
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\end{isamarkuptext}% |
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% |
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\begin{isamarkuptext}% |
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\begin{isabelle}% |
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\ \ \ \ \ {\isasymlbrakk}i\ {\isasymle}\ j{\isacharsemicolon}\ k\ {\isasymle}\ l{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharasterisk}\ k\ {\isasymle}\ j\ {\isacharasterisk}\ l% |
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\end{isabelle} |
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\rulename{mult_le_mono} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isasymlbrakk}i\ {\isacharless}\ j{\isacharsemicolon}\ {\isadigit{0}}\ {\isacharless}\ k{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharasterisk}\ k\ {\isacharless}\ j\ {\isacharasterisk}\ k% |
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\end{isabelle} |
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\rulename{mult_less_mono1} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\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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\ \ \ \ \ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharplus}\ n\ {\isacharasterisk}\ k% |
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\end{isabelle} |
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\rulename{add_mult_distrib} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharminus}\ n\ {\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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\ \ \ \ \ m\ mod\ n\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ mod\ {\isacharparenleft}n\ {\isacharasterisk}\ k{\isacharparenright}% |
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\end{isabelle} |
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\rulename{mod_mult_distrib} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ P\ {\isacharparenleft}a\ {\isacharminus}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenleft}a\ {\isacharless}\ b\ {\isasymlongrightarrow}\ P\ {\isadigit{0}}{\isacharparenright}\ {\isasymand}\ {\isacharparenleft}{\isasymforall}d{\isachardot}\ a\ {\isacharequal}\ b\ {\isacharplus}\ d\ {\isasymlongrightarrow}\ P\ d{\isacharparenright}{\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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\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}n{\isacharminus}{\isadigit{1}}{\isacharparenright}{\isacharasterisk}{\isacharparenleft}n{\isacharplus}{\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ n{\isacharasterisk}n\ {\isacharminus}\ {\isadigit{1}}{\isachardoublequote}\isanewline |
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\isacommand{apply}\ {\isacharparenleft}simp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharparenright}\isanewline |
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\isacommand{done}% |
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\begin{isamarkuptext}% |
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\begin{isabelle}% |
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\ \ \ \ \ m\ mod\ n\ {\isacharequal}\ {\isacharparenleft}if\ m\ {\isacharless}\ n\ then\ m\ else\ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ mod\ n{\isacharparenright}% |
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\end{isabelle} |
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\rulename{mod_if} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ m\ div\ n\ {\isacharasterisk}\ n\ {\isacharplus}\ m\ mod\ n\ {\isacharequal}\ m% |
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\end{isabelle} |
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\rulename{mod_div_equality} |
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|
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c% |
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\end{isabelle} |
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\rulename{div_mult1_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
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\end{isabelle} |
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\rulename{mod_mult1_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
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\end{isabelle} |
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\rulename{div_mult2_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b% |
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\end{isabelle} |
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\rulename{mod_mult2_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ c\ {\isacharasterisk}\ a\ div\ {\isacharparenleft}c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ a\ div\ b% |
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\end{isabelle} |
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\rulename{div_mult_mult1} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ div\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}% |
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\end{isabelle} |
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\rulename{DIVISION_BY_ZERO_DIV} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ mod\ {\isadigit{0}}\ {\isacharequal}\ a% |
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\end{isabelle} |
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\rulename{DIVISION_BY_ZERO_MOD} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isasymlbrakk}m\ dvd\ n{\isacharsemicolon}\ n\ dvd\ m{\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n% |
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\end{isabelle} |
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\rulename{dvd_anti_sym} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isasymlbrakk}k\ dvd\ m{\isacharsemicolon}\ k\ dvd\ n{\isasymrbrakk}\ {\isasymLongrightarrow}\ k\ dvd\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}% |
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\end{isabelle} |
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\rulename{dvd_add} |
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|
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For the integers, I'd list a few theorems that somehow involve negative |
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numbers. |
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|
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Division, remainder of negatives |
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isacharhash}{\isadigit{0}}\ {\isasymle}\ a\ mod\ b% |
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\end{isabelle} |
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\rulename{pos_mod_sign} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b% |
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\end{isabelle} |
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\rulename{pos_mod_bound} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ b\ {\isacharless}\ {\isacharhash}{\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ {\isacharhash}{\isadigit{0}}% |
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\end{isabelle} |
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\rulename{neg_mod_sign} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ b\ {\isacharless}\ {\isacharhash}{\isadigit{0}}\ {\isasymLongrightarrow}\ b\ {\isacharless}\ a\ mod\ b% |
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\end{isabelle} |
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\rulename{neg_mod_bound} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ div\ c\ {\isacharequal}\isanewline |
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\ \ \ \ \ a\ div\ c\ {\isacharplus}\ b\ div\ c\ {\isacharplus}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ div\ c% |
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\end{isabelle} |
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\rulename{zdiv_zadd1_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ mod\ c\ {\isacharequal}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ mod\ c% |
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\end{isabelle} |
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\rulename{zmod_zadd1_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c% |
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\end{isabelle} |
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\rulename{zdiv_zmult1_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
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\end{isabelle} |
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\rulename{zmod_zmult1_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
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\end{isabelle} |
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\rulename{zdiv_zmult2_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isacharhash}{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b% |
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\end{isabelle} |
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\rulename{zmod_zmult2_eq} |
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|
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\begin{isabelle}% |
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\ \ \ \ \ {\isasymbar}x\ {\isacharasterisk}\ y{\isasymbar}\ {\isacharequal}\ {\isasymbar}x{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}y{\isasymbar}% |
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\end{isabelle} |
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\rulename{abs_mult}% |
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\end{isamarkuptext}% |
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\isacommand{end}\isanewline |
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\end{isabellebody}% |
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%%% Local Variables: |
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%%% mode: latex |
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%%% TeX-master: "root" |
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%%% End: |