author | haftmann |
Fri, 27 Jun 2008 09:55:02 +0200 | |
changeset 27376 | ffe9b958bada |
parent 25056 | 743f3603ba8b |
child 27658 | 674496eb5965 |
permissions | -rw-r--r-- |
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\begin{isabellebody}% |
3 |
\def\isabellecontext{Numbers}% |
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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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\isatagtheory |
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\isacommand{theory}\isamarkupfalse% |
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\ Numbers\isanewline |
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\isakeyword{imports}\ Complex{\isacharunderscore}Main\isanewline |
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14 |
\isakeyword{begin}% |
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\endisatagtheory |
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{\isafoldtheory}% |
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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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\ {\isachardoublequoteopen}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequoteclose}\isanewline |
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\isacommand{ML}\isamarkupfalse% |
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\ {\isachardoublequoteopen}ThyOutput{\isachardot}indent\ {\isacharcolon}{\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}% |
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\endisatagML |
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{\isafoldML}% |
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% |
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\isadelimML |
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% |
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\endisadelimML |
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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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45 |
\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\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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% |
52 |
\begin{isamarkuptxt}% |
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\begin{isabelle}% |
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54 |
\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m% |
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55 |
\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 |
61 |
{\isafoldproof}% |
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62 |
% |
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63 |
\isadelimproof |
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64 |
% |
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65 |
\endisadelimproof |
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66 |
\isanewline |
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15481 | 67 |
\isanewline |
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\isacommand{consts}\isamarkupfalse% |
69 |
\ h\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline |
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\isacommand{recdef}\isamarkupfalse% |
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71 |
\ h\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}\isanewline |
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{\isachardoublequoteopen}h\ i\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isacharequal}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequoteclose}% |
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\begin{isamarkuptext}% |
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\isa{h\ {\isadigit{3}}\ {\isacharequal}\ {\isadigit{2}}} |
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\isa{h\ i\ {\isacharequal}\ i}% |
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\end{isamarkuptext}% |
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\isamarkuptrue% |
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% |
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\begin{isamarkuptext}% |
80 |
\begin{isabelle}% |
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Numeral{\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}% |
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\end{isabelle} |
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\rulename{numeral_0_eq_0} |
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\begin{isabelle}% |
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86 |
Numeral{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}{\isasymColon}{\isacharprime}a{\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}}\ {\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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\begin{isabelle}% |
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n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}% |
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\end{isabelle} |
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\rulename{add_2_eq_Suc'} |
99 |
||
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\begin{isabelle}% |
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a\ {\isacharplus}\ b\ {\isacharplus}\ c\ {\isacharequal}\ a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\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\ {\isacharplus}\ b\ {\isacharequal}\ b\ {\isacharplus}\ a% |
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\end{isabelle} |
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\rulename{add_commute} |
10602 | 109 |
|
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\begin{isabelle}% |
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a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharplus}\ {\isacharparenleft}a\ {\isacharplus}\ c{\isacharparenright}% |
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\end{isabelle} |
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\rulename{add_left_commute} |
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115 |
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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\ {\isachardoublequoteopen}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}{\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}}{\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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\end{isabelle}% |
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\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isacharparenleft}simp\ add{\isacharcolon}\ add{\isacharunderscore}ac\ mult{\isacharunderscore}ac{\isacharparenright}% |
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\begin{isamarkuptxt}% |
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\begin{isabelle}% |
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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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\isaindent{\ {\isadigit{1}}{\isachardot}\ }f\ {\isacharparenleft}i\ {\isacharplus}\ {\isacharparenleft}m\ {\isacharasterisk}\ n\ {\isacharplus}\ j\ {\isacharasterisk}\ {\isacharparenleft}k\ {\isacharasterisk}\ l{\isacharparenright}{\isacharparenright}{\isacharparenright}% |
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138 |
\end{isabelle}% |
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139 |
\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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152 |
m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\isasymle}\ n\ div\ k% |
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153 |
\end{isabelle} |
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\rulename{div_le_mono} |
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\begin{isabelle}% |
157 |
{\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharminus}\ n\ {\isacharasterisk}\ k% |
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158 |
\end{isabelle} |
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\rulename{diff_mult_distrib} |
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\begin{isabelle}% |
162 |
m\ mod\ n\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ mod\ {\isacharparenleft}n\ {\isacharasterisk}\ k{\isacharparenright}% |
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163 |
\end{isabelle} |
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\rulename{mod_mult_distrib} |
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\begin{isabelle}% |
167 |
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}% |
170 |
\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
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\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}\ {\isacharequal}\ n\ {\isacharasterisk}\ n\ {\isacharminus}\ {\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\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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\ {\isacharparenleft}clarsimp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split\ iff\ del{\isacharcolon}\ less{\isacharunderscore}Suc{\isadigit{0}}{\isacharparenright}\isanewline |
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\ % |
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\isamarkupcmt{\begin{isabelle}% |
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184 |
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}d{\isachardot}\ {\isasymlbrakk}n\ {\isacharless}\ Suc\ {\isadigit{0}}{\isacharsemicolon}\ n\ {\isacharasterisk}\ n\ {\isacharequal}\ Suc\ d{\isasymrbrakk}\ {\isasymLongrightarrow}\ d\ {\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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\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequoteopen}n{\isacharequal}{\isadigit{0}}{\isachardoublequoteclose}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline |
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\isacommand{done}\isamarkupfalse% |
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191 |
% |
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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 |
d2758965362e
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200 |
\isanewline |
17175 | 201 |
\isacommand{lemma}\isamarkupfalse% |
202 |
\ {\isachardoublequoteopen}{\isacharparenleft}n\ {\isacharminus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}n\ {\isacharplus}\ {\isadigit{2}}{\isacharparenright}\ {\isacharequal}\ n\ {\isacharasterisk}\ n\ {\isacharminus}\ {\isacharparenleft}{\isadigit{4}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isachardoublequoteclose}\isanewline |
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% |
204 |
\isadelimproof |
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% |
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206 |
\endisadelimproof |
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207 |
% |
|
208 |
\isatagproof |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isacharparenleft}simp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharcomma}\ clarify{\isacharparenright}\isanewline |
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\ % |
212 |
\isamarkupcmt{\begin{isabelle}% |
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213 |
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}d{\isachardot}\ {\isasymlbrakk}n\ {\isacharless}\ {\isadigit{2}}{\isacharsemicolon}\ n\ {\isacharasterisk}\ n\ {\isacharequal}\ {\isadigit{4}}\ {\isacharplus}\ d{\isasymrbrakk}\ {\isasymLongrightarrow}\ d\ {\isacharequal}\ {\isadigit{0}}% |
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214 |
\end{isabelle}% |
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215 |
} |
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216 |
\isanewline |
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\isacommand{apply}\isamarkupfalse% |
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\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequoteopen}n{\isacharequal}{\isadigit{0}}\ {\isacharbar}\ n{\isacharequal}{\isadigit{1}}{\isachardoublequoteclose}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline |
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219 |
\isacommand{done}\isamarkupfalse% |
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220 |
% |
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17056 | 221 |
\endisatagproof |
222 |
{\isafoldproof}% |
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223 |
% |
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224 |
\isadelimproof |
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225 |
% |
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226 |
\endisadelimproof |
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11866 | 227 |
% |
11187 | 228 |
\begin{isamarkuptext}% |
229 |
\begin{isabelle}% |
|
230 |
m\ mod\ n\ {\isacharequal}\ {\isacharparenleft}if\ m\ {\isacharless}\ n\ then\ m\ else\ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ mod\ n{\isacharparenright}% |
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231 |
\end{isabelle} |
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11174 | 232 |
\rulename{mod_if} |
11187 | 233 |
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234 |
\begin{isabelle}% |
|
235 |
m\ div\ n\ {\isacharasterisk}\ n\ {\isacharplus}\ m\ mod\ n\ {\isacharequal}\ m% |
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10602 | 236 |
\end{isabelle} |
11174 | 237 |
\rulename{mod_div_equality} |
10602 | 238 |
|
239 |
||
11187 | 240 |
\begin{isabelle}% |
241 |
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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242 |
\end{isabelle} |
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11174 | 243 |
\rulename{div_mult1_eq} |
10602 | 244 |
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11187 | 245 |
\begin{isabelle}% |
246 |
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
|
247 |
\end{isabelle} |
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11174 | 248 |
\rulename{mod_mult1_eq} |
10602 | 249 |
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11187 | 250 |
\begin{isabelle}% |
251 |
a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
|
252 |
\end{isabelle} |
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11174 | 253 |
\rulename{div_mult2_eq} |
10602 | 254 |
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11187 | 255 |
\begin{isabelle}% |
256 |
a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b% |
|
257 |
\end{isabelle} |
|
11174 | 258 |
\rulename{mod_mult2_eq} |
10602 | 259 |
|
11187 | 260 |
\begin{isabelle}% |
261 |
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ c\ {\isacharasterisk}\ a\ div\ {\isacharparenleft}c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ a\ div\ b% |
|
262 |
\end{isabelle} |
|
11174 | 263 |
\rulename{div_mult_mult1} |
10602 | 264 |
|
11187 | 265 |
\begin{isabelle}% |
266 |
a\ div\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}% |
|
267 |
\end{isabelle} |
|
11174 | 268 |
\rulename{DIVISION_BY_ZERO_DIV} |
10602 | 269 |
|
11187 | 270 |
\begin{isabelle}% |
271 |
a\ mod\ {\isadigit{0}}\ {\isacharequal}\ a% |
|
272 |
\end{isabelle} |
|
11174 | 273 |
\rulename{DIVISION_BY_ZERO_MOD} |
10602 | 274 |
|
11187 | 275 |
\begin{isabelle}% |
276 |
{\isasymlbrakk}m\ dvd\ n{\isacharsemicolon}\ n\ dvd\ m{\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n% |
|
277 |
\end{isabelle} |
|
11174 | 278 |
\rulename{dvd_anti_sym} |
10602 | 279 |
|
11187 | 280 |
\begin{isabelle}% |
281 |
{\isasymlbrakk}k\ dvd\ m{\isacharsemicolon}\ k\ dvd\ n{\isasymrbrakk}\ {\isasymLongrightarrow}\ k\ dvd\ m\ {\isacharplus}\ n% |
|
282 |
\end{isabelle} |
|
11174 | 283 |
\rulename{dvd_add} |
10602 | 284 |
|
285 |
For the integers, I'd list a few theorems that somehow involve negative |
|
13758 | 286 |
numbers.% |
287 |
\end{isamarkuptext}% |
|
288 |
\isamarkuptrue% |
|
289 |
% |
|
290 |
\begin{isamarkuptext}% |
|
10602 | 291 |
Division, remainder of negatives |
292 |
||
293 |
||
11187 | 294 |
\begin{isabelle}% |
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295 |
{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ a\ mod\ b% |
11187 | 296 |
\end{isabelle} |
11174 | 297 |
\rulename{pos_mod_sign} |
10602 | 298 |
|
11187 | 299 |
\begin{isabelle}% |
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|
300 |
{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b% |
11174 | 301 |
\end{isabelle} |
11187 | 302 |
\rulename{pos_mod_bound} |
10602 | 303 |
|
11187 | 304 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
305 |
b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ {\isadigit{0}}% |
11174 | 306 |
\end{isabelle} |
11187 | 307 |
\rulename{neg_mod_sign} |
10602 | 308 |
|
11187 | 309 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
310 |
b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ b\ {\isacharless}\ a\ mod\ b% |
11187 | 311 |
\end{isabelle} |
11174 | 312 |
\rulename{neg_mod_bound} |
10602 | 313 |
|
11187 | 314 |
\begin{isabelle}% |
315 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ div\ c\ {\isacharequal}\ a\ div\ c\ {\isacharplus}\ b\ div\ c\ {\isacharplus}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ div\ c% |
|
11174 | 316 |
\end{isabelle} |
11187 | 317 |
\rulename{zdiv_zadd1_eq} |
10602 | 318 |
|
11187 | 319 |
\begin{isabelle}% |
320 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ mod\ c\ {\isacharequal}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ mod\ c% |
|
321 |
\end{isabelle} |
|
10602 | 322 |
\rulename{zmod_zadd1_eq} |
323 |
||
11187 | 324 |
\begin{isabelle}% |
325 |
a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c% |
|
10602 | 326 |
\end{isabelle} |
11187 | 327 |
\rulename{zdiv_zmult1_eq} |
10602 | 328 |
|
11187 | 329 |
\begin{isabelle}% |
330 |
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
|
331 |
\end{isabelle} |
|
10602 | 332 |
\rulename{zmod_zmult1_eq} |
333 |
||
11187 | 334 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
335 |
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
11174 | 336 |
\end{isabelle} |
11187 | 337 |
\rulename{zdiv_zmult2_eq} |
10602 | 338 |
|
11187 | 339 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
340 |
{\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% |
11187 | 341 |
\end{isabelle} |
14400 | 342 |
\rulename{zmod_zmult2_eq}% |
11187 | 343 |
\end{isamarkuptext}% |
17175 | 344 |
\isamarkuptrue% |
345 |
\isacommand{lemma}\isamarkupfalse% |
|
346 |
\ {\isachardoublequoteopen}abs\ {\isacharparenleft}x{\isacharplus}y{\isacharparenright}\ {\isasymle}\ abs\ x\ {\isacharplus}\ abs\ {\isacharparenleft}y\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
17056 | 347 |
% |
348 |
\isadelimproof |
|
349 |
% |
|
350 |
\endisadelimproof |
|
351 |
% |
|
352 |
\isatagproof |
|
17175 | 353 |
\isacommand{by}\isamarkupfalse% |
354 |
\ arith% |
|
17056 | 355 |
\endisatagproof |
356 |
{\isafoldproof}% |
|
357 |
% |
|
358 |
\isadelimproof |
|
359 |
\isanewline |
|
360 |
% |
|
361 |
\endisadelimproof |
|
15481 | 362 |
\isanewline |
17175 | 363 |
\isacommand{lemma}\isamarkupfalse% |
364 |
\ {\isachardoublequoteopen}abs\ {\isacharparenleft}{\isadigit{2}}{\isacharasterisk}x{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ abs\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
17056 | 365 |
% |
366 |
\isadelimproof |
|
367 |
% |
|
368 |
\endisadelimproof |
|
369 |
% |
|
370 |
\isatagproof |
|
17175 | 371 |
\isacommand{by}\isamarkupfalse% |
372 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ abs{\isacharunderscore}if{\isacharparenright}% |
|
17056 | 373 |
\endisatagproof |
374 |
{\isafoldproof}% |
|
375 |
% |
|
376 |
\isadelimproof |
|
377 |
% |
|
378 |
\endisadelimproof |
|
11866 | 379 |
% |
11187 | 380 |
\begin{isamarkuptext}% |
13758 | 381 |
Induction rules for the Integers |
382 |
||
383 |
\begin{isabelle}% |
|
384 |
{\isasymlbrakk}k\ {\isasymle}\ i{\isacharsemicolon}\ P\ k{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}k\ {\isasymle}\ i{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i% |
|
385 |
\end{isabelle} |
|
386 |
\rulename{int_ge_induct} |
|
387 |
||
388 |
\begin{isabelle}% |
|
389 |
{\isasymlbrakk}k\ {\isacharless}\ i{\isacharsemicolon}\ P\ {\isacharparenleft}k\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}k\ {\isacharless}\ i{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharplus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i% |
|
390 |
\end{isabelle} |
|
391 |
\rulename{int_gr_induct} |
|
392 |
||
393 |
\begin{isabelle}% |
|
394 |
{\isasymlbrakk}i\ {\isasymle}\ k{\isacharsemicolon}\ P\ k{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isasymle}\ k{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i% |
|
395 |
\end{isabelle} |
|
396 |
\rulename{int_le_induct} |
|
397 |
||
398 |
\begin{isabelle}% |
|
399 |
{\isasymlbrakk}i\ {\isacharless}\ k{\isacharsemicolon}\ P\ {\isacharparenleft}k\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isacharsemicolon}\ {\isasymAnd}i{\isachardot}\ {\isasymlbrakk}i\ {\isacharless}\ k{\isacharsemicolon}\ P\ i{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}i\ {\isacharminus}\ {\isadigit{1}}{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ i% |
|
400 |
\end{isabelle} |
|
401 |
\rulename{int_less_induct}% |
|
402 |
\end{isamarkuptext}% |
|
403 |
\isamarkuptrue% |
|
404 |
% |
|
405 |
\begin{isamarkuptext}% |
|
14400 | 406 |
FIELDS |
10776 | 407 |
|
11187 | 408 |
\begin{isabelle}% |
25056 | 409 |
x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ {\isasymexists}z{\isachargreater}x{\isachardot}\ z\ {\isacharless}\ y% |
11187 | 410 |
\end{isabelle} |
14295 | 411 |
\rulename{dense} |
10776 | 412 |
|
11187 | 413 |
\begin{isabelle}% |
14288 | 414 |
a\ {\isacharasterisk}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ b\ {\isacharslash}\ c% |
11187 | 415 |
\end{isabelle} |
14288 | 416 |
\rulename{times_divide_eq_right} |
10776 | 417 |
|
11187 | 418 |
\begin{isabelle}% |
14288 | 419 |
b\ {\isacharslash}\ c\ {\isacharasterisk}\ a\ {\isacharequal}\ b\ {\isacharasterisk}\ a\ {\isacharslash}\ c% |
11187 | 420 |
\end{isabelle} |
14288 | 421 |
\rulename{times_divide_eq_left} |
10776 | 422 |
|
11187 | 423 |
\begin{isabelle}% |
14288 | 424 |
a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ c\ {\isacharslash}\ b% |
11187 | 425 |
\end{isabelle} |
14288 | 426 |
\rulename{divide_divide_eq_right} |
10776 | 427 |
|
11187 | 428 |
\begin{isabelle}% |
14288 | 429 |
a\ {\isacharslash}\ b\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}% |
11187 | 430 |
\end{isabelle} |
14288 | 431 |
\rulename{divide_divide_eq_left} |
10776 | 432 |
|
11187 | 433 |
\begin{isabelle}% |
14295 | 434 |
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharminus}\ a\ {\isacharslash}\ b% |
11187 | 435 |
\end{isabelle} |
14295 | 436 |
\rulename{minus_divide_left} |
10776 | 437 |
|
11187 | 438 |
\begin{isabelle}% |
14295 | 439 |
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharminus}\ b% |
11187 | 440 |
\end{isabelle} |
14295 | 441 |
\rulename{minus_divide_right} |
10776 | 442 |
|
443 |
This last NOT a simprule |
|
444 |
||
11187 | 445 |
\begin{isabelle}% |
14295 | 446 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ c\ {\isacharplus}\ b\ {\isacharslash}\ c% |
11174 | 447 |
\end{isabelle} |
14295 | 448 |
\rulename{add_divide_distrib}% |
11187 | 449 |
\end{isamarkuptext}% |
17175 | 450 |
\isamarkuptrue% |
451 |
\isacommand{lemma}\isamarkupfalse% |
|
452 |
\ {\isachardoublequoteopen}{\isadigit{3}}{\isacharslash}{\isadigit{4}}\ {\isacharless}\ {\isacharparenleft}{\isadigit{7}}{\isacharslash}{\isadigit{8}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
17056 | 453 |
% |
454 |
\isadelimproof |
|
455 |
% |
|
456 |
\endisadelimproof |
|
457 |
% |
|
458 |
\isatagproof |
|
17175 | 459 |
\isacommand{by}\isamarkupfalse% |
460 |
\ simp% |
|
17056 | 461 |
\endisatagproof |
462 |
{\isafoldproof}% |
|
463 |
% |
|
464 |
\isadelimproof |
|
465 |
\ \isanewline |
|
466 |
% |
|
467 |
\endisadelimproof |
|
11174 | 468 |
\isanewline |
17175 | 469 |
\isacommand{lemma}\isamarkupfalse% |
470 |
\ {\isachardoublequoteopen}P\ {\isacharparenleft}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isacharparenright}{\isachardoublequoteclose}% |
|
17056 | 471 |
\isadelimproof |
472 |
% |
|
473 |
\endisadelimproof |
|
474 |
% |
|
475 |
\isatagproof |
|
16353 | 476 |
% |
477 |
\begin{isamarkuptxt}% |
|
478 |
\begin{isabelle}% |
|
479 |
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\isacharparenright}% |
|
480 |
\end{isabelle}% |
|
481 |
\end{isamarkuptxt}% |
|
17175 | 482 |
\isamarkuptrue% |
483 |
\isacommand{apply}\isamarkupfalse% |
|
484 |
\ simp% |
|
16353 | 485 |
\begin{isamarkuptxt}% |
486 |
\begin{isabelle}% |
|
487 |
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{2}}\ {\isacharslash}\ {\isadigit{5}}{\isacharparenright}% |
|
488 |
\end{isabelle}% |
|
489 |
\end{isamarkuptxt}% |
|
17175 | 490 |
\isamarkuptrue% |
491 |
\isacommand{oops}\isamarkupfalse% |
|
492 |
% |
|
17056 | 493 |
\endisatagproof |
494 |
{\isafoldproof}% |
|
495 |
% |
|
496 |
\isadelimproof |
|
497 |
% |
|
498 |
\endisadelimproof |
|
499 |
\isanewline |
|
15481 | 500 |
\isanewline |
17175 | 501 |
\isacommand{lemma}\isamarkupfalse% |
502 |
\ {\isachardoublequoteopen}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequoteclose}% |
|
17056 | 503 |
\isadelimproof |
504 |
% |
|
505 |
\endisadelimproof |
|
506 |
% |
|
507 |
\isatagproof |
|
16353 | 508 |
% |
509 |
\begin{isamarkuptxt}% |
|
510 |
\begin{isabelle}% |
|
511 |
\ {\isadigit{1}}{\isachardot}\ {\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ x% |
|
512 |
\end{isabelle}% |
|
513 |
\end{isamarkuptxt}% |
|
17175 | 514 |
\isamarkuptrue% |
515 |
\isacommand{apply}\isamarkupfalse% |
|
516 |
\ simp% |
|
16353 | 517 |
\begin{isamarkuptxt}% |
518 |
\begin{isabelle}% |
|
519 |
\ {\isadigit{1}}{\isachardot}\ {\isadigit{2}}\ {\isacharless}\ x\ {\isacharasterisk}\ {\isadigit{5}}% |
|
520 |
\end{isabelle}% |
|
521 |
\end{isamarkuptxt}% |
|
17175 | 522 |
\isamarkuptrue% |
523 |
\isacommand{oops}\isamarkupfalse% |
|
524 |
% |
|
17056 | 525 |
\endisatagproof |
526 |
{\isafoldproof}% |
|
527 |
% |
|
528 |
\isadelimproof |
|
529 |
% |
|
530 |
\endisadelimproof |
|
14400 | 531 |
% |
532 |
\begin{isamarkuptext}% |
|
533 |
Ring and Field |
|
534 |
||
535 |
Requires a field, or else an ordered ring |
|
536 |
||
537 |
\begin{isabelle}% |
|
538 |
{\isacharparenleft}a\ {\isacharasterisk}\ b\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isasymor}\ b\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}{\isacharparenright}% |
|
539 |
\end{isabelle} |
|
540 |
\rulename{mult_eq_0_iff} |
|
541 |
||
542 |
\begin{isabelle}% |
|
543 |
{\isacharparenleft}a\ {\isacharasterisk}\ c\ {\isacharequal}\ b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}% |
|
544 |
\end{isabelle} |
|
545 |
\rulename{mult_cancel_right} |
|
546 |
||
547 |
\begin{isabelle}% |
|
23504 | 548 |
{\isacharparenleft}c\ {\isacharasterisk}\ a\ {\isacharequal}\ c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}% |
14400 | 549 |
\end{isabelle} |
23504 | 550 |
\rulename{mult_cancel_left}% |
14400 | 551 |
\end{isamarkuptext}% |
17175 | 552 |
\isamarkuptrue% |
17056 | 553 |
% |
554 |
\isadelimML |
|
555 |
% |
|
556 |
\endisadelimML |
|
557 |
% |
|
558 |
\isatagML |
|
17175 | 559 |
\isacommand{ML}\isamarkupfalse% |
17181 | 560 |
{\isacharverbatimopen}set\ show{\isacharunderscore}sorts{\isacharverbatimclose}% |
17056 | 561 |
\endisatagML |
562 |
{\isafoldML}% |
|
563 |
% |
|
564 |
\isadelimML |
|
565 |
% |
|
566 |
\endisadelimML |
|
14400 | 567 |
% |
568 |
\begin{isamarkuptext}% |
|
569 |
effect of show sorts on the above |
|
570 |
||
571 |
\begin{isabelle}% |
|
23504 | 572 |
{\isacharparenleft}{\isacharparenleft}c{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}a{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}\ {\isacharequal}\isanewline |
573 |
\isaindent{{\isacharparenleft}}c\ {\isacharasterisk}\ {\isacharparenleft}b{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}{\isacharparenright}\ {\isacharequal}\isanewline |
|
574 |
{\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isasymColon}ring{\isacharunderscore}no{\isacharunderscore}zero{\isacharunderscore}divisors{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}% |
|
14400 | 575 |
\end{isabelle} |
23504 | 576 |
\rulename{mult_cancel_left}% |
14400 | 577 |
\end{isamarkuptext}% |
17175 | 578 |
\isamarkuptrue% |
17056 | 579 |
% |
580 |
\isadelimML |
|
581 |
% |
|
582 |
\endisadelimML |
|
583 |
% |
|
584 |
\isatagML |
|
17175 | 585 |
\isacommand{ML}\isamarkupfalse% |
17181 | 586 |
{\isacharverbatimopen}reset\ show{\isacharunderscore}sorts{\isacharverbatimclose}% |
17056 | 587 |
\endisatagML |
588 |
{\isafoldML}% |
|
589 |
% |
|
590 |
\isadelimML |
|
591 |
% |
|
592 |
\endisadelimML |
|
14400 | 593 |
% |
594 |
\begin{isamarkuptext}% |
|
595 |
absolute value |
|
596 |
||
597 |
\begin{isabelle}% |
|
598 |
{\isasymbar}a\ {\isacharasterisk}\ b{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}b{\isasymbar}% |
|
599 |
\end{isabelle} |
|
600 |
\rulename{abs_mult} |
|
601 |
||
602 |
\begin{isabelle}% |
|
603 |
{\isacharparenleft}{\isasymbar}a{\isasymbar}\ {\isasymle}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isasymle}\ b\ {\isasymand}\ {\isacharminus}\ a\ {\isasymle}\ b{\isacharparenright}% |
|
604 |
\end{isabelle} |
|
605 |
\rulename{abs_le_iff} |
|
606 |
||
607 |
\begin{isabelle}% |
|
608 |
{\isasymbar}a\ {\isacharplus}\ b{\isasymbar}\ {\isasymle}\ {\isasymbar}a{\isasymbar}\ {\isacharplus}\ {\isasymbar}b{\isasymbar}% |
|
609 |
\end{isabelle} |
|
610 |
\rulename{abs_triangle_ineq} |
|
611 |
||
612 |
\begin{isabelle}% |
|
613 |
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharequal}\ a\ {\isacharcircum}\ m\ {\isacharasterisk}\ a\ {\isacharcircum}\ n% |
|
614 |
\end{isabelle} |
|
615 |
\rulename{power_add} |
|
616 |
||
617 |
\begin{isabelle}% |
|
618 |
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isacharcircum}\ m{\isacharparenright}\ {\isacharcircum}\ n% |
|
619 |
\end{isabelle} |
|
620 |
\rulename{power_mult} |
|
621 |
||
622 |
\begin{isabelle}% |
|
623 |
{\isasymbar}a\ {\isacharcircum}\ n{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharcircum}\ n% |
|
624 |
\end{isabelle} |
|
625 |
\rulename{power_abs}% |
|
626 |
\end{isamarkuptext}% |
|
17175 | 627 |
\isamarkuptrue% |
17056 | 628 |
% |
629 |
\isadelimtheory |
|
630 |
% |
|
631 |
\endisadelimtheory |
|
632 |
% |
|
633 |
\isatagtheory |
|
17175 | 634 |
\isacommand{end}\isamarkupfalse% |
635 |
% |
|
17056 | 636 |
\endisatagtheory |
637 |
{\isafoldtheory}% |
|
638 |
% |
|
639 |
\isadelimtheory |
|
640 |
% |
|
641 |
\endisadelimtheory |
|
642 |
\isanewline |
|
11187 | 643 |
\end{isabellebody}% |
10602 | 644 |
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|
646 |
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647 |
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