author | nipkow |
Tue, 03 Mar 2009 17:05:18 +0100 | |
changeset 30224 | 79136ce06bdb |
parent 30200 | 0db3a35eab01 |
child 30242 | aea5d7fa7ef5 |
permissions | -rw-r--r-- |
10602 | 1 |
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\begin{isabellebody}% |
3 |
\def\isabellecontext{Numbers}% |
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17056 | 4 |
% |
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\isadelimtheory |
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\endisadelimtheory |
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8 |
% |
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9 |
\isatagtheory |
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17175 | 10 |
\isacommand{theory}\isamarkupfalse% |
27376 | 11 |
\ Numbers\isanewline |
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\isakeyword{imports}\ Complex{\isacharunderscore}Main\isanewline |
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13 |
\isakeyword{begin}% |
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17056 | 14 |
\endisatagtheory |
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{\isafoldtheory}% |
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% |
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\isadelimtheory |
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10602 | 18 |
\isanewline |
17056 | 19 |
% |
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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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17175 | 28 |
\isacommand{ML}\isamarkupfalse% |
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\ {\isachardoublequoteopen}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequoteclose}\isanewline |
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30 |
\isacommand{ML}\isamarkupfalse% |
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22649 | 31 |
\ {\isachardoublequoteopen}ThyOutput{\isachardot}indent\ {\isacharcolon}{\isacharequal}\ {\isadigit{0}}{\isachardoublequoteclose}% |
17056 | 32 |
\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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% |
11187 | 39 |
\begin{isamarkuptext}% |
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numeric literals; default simprules; can re-orient% |
11187 | 41 |
\end{isamarkuptext}% |
17175 | 42 |
\isamarkuptrue% |
43 |
\isacommand{lemma}\isamarkupfalse% |
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44 |
\ {\isachardoublequoteopen}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequoteclose}% |
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17056 | 45 |
\isadelimproof |
46 |
% |
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47 |
\endisadelimproof |
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48 |
% |
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49 |
\isatagproof |
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16353 | 50 |
% |
51 |
\begin{isamarkuptxt}% |
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52 |
\begin{isabelle}% |
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53 |
\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m% |
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54 |
\end{isabelle}% |
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55 |
\end{isamarkuptxt}% |
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17175 | 56 |
\isamarkuptrue% |
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\isacommand{oops}\isamarkupfalse% |
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58 |
% |
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17056 | 59 |
\endisatagproof |
60 |
{\isafoldproof}% |
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61 |
% |
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62 |
\isadelimproof |
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63 |
% |
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64 |
\endisadelimproof |
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65 |
\isanewline |
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15481 | 66 |
\isanewline |
17175 | 67 |
\isacommand{consts}\isamarkupfalse% |
68 |
\ h\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequoteopen}nat\ {\isasymRightarrow}\ nat{\isachardoublequoteclose}\isanewline |
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69 |
\isacommand{recdef}\isamarkupfalse% |
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70 |
\ h\ {\isachardoublequoteopen}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequoteclose}\isanewline |
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71 |
{\isachardoublequoteopen}h\ i\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isacharequal}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequoteclose}% |
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11187 | 72 |
\begin{isamarkuptext}% |
11708 | 73 |
\isa{h\ {\isadigit{3}}\ {\isacharequal}\ {\isadigit{2}}} |
11187 | 74 |
\isa{h\ i\ {\isacharequal}\ i}% |
75 |
\end{isamarkuptext}% |
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\isamarkuptrue% |
10878 | 77 |
% |
11187 | 78 |
\begin{isamarkuptext}% |
79 |
\begin{isabelle}% |
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80 |
Numeral{\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}% |
11187 | 81 |
\end{isabelle} |
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\rulename{numeral_0_eq_0} |
10602 | 83 |
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11187 | 84 |
\begin{isabelle}% |
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85 |
Numeral{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}{\isasymColon}{\isacharprime}a{\isacharparenright}% |
10602 | 86 |
\end{isabelle} |
11187 | 87 |
\rulename{numeral_1_eq_1} |
11174 | 88 |
|
11187 | 89 |
\begin{isabelle}% |
11708 | 90 |
{\isadigit{2}}\ {\isacharplus}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}% |
11187 | 91 |
\end{isabelle} |
10602 | 92 |
\rulename{add_2_eq_Suc} |
11187 | 93 |
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94 |
\begin{isabelle}% |
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11708 | 95 |
n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}% |
10602 | 96 |
\end{isabelle} |
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\rulename{add_2_eq_Suc'} |
98 |
||
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\begin{isabelle}% |
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a\ {\isacharplus}\ b\ {\isacharplus}\ c\ {\isacharequal}\ a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}% |
11174 | 101 |
\end{isabelle} |
11187 | 102 |
\rulename{add_assoc} |
10602 | 103 |
|
11187 | 104 |
\begin{isabelle}% |
14270 | 105 |
a\ {\isacharplus}\ b\ {\isacharequal}\ b\ {\isacharplus}\ a% |
11174 | 106 |
\end{isabelle} |
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\rulename{add_commute} |
10602 | 108 |
|
11187 | 109 |
\begin{isabelle}% |
14270 | 110 |
a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharplus}\ {\isacharparenleft}a\ {\isacharplus}\ c{\isacharparenright}% |
11187 | 111 |
\end{isabelle} |
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\rulename{add_left_commute} |
10602 | 113 |
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114 |
these form add_ac; similarly there is mult_ac% |
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11187 | 115 |
\end{isamarkuptext}% |
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\isamarkuptrue% |
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\isacommand{lemma}\isamarkupfalse% |
|
118 |
\ {\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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17056 | 119 |
\isadelimproof |
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% |
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\endisadelimproof |
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122 |
% |
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123 |
\isatagproof |
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16353 | 124 |
% |
125 |
\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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128 |
\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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137 |
\end{isabelle}% |
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138 |
\end{isamarkuptxt}% |
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\isamarkuptrue% |
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\isacommand{oops}\isamarkupfalse% |
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% |
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17056 | 142 |
\endisatagproof |
143 |
{\isafoldproof}% |
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144 |
% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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\begin{isamarkuptext}% |
150 |
\begin{isabelle}% |
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151 |
m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\isasymle}\ n\ div\ k% |
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152 |
\end{isabelle} |
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\rulename{div_le_mono} |
10602 | 154 |
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11187 | 155 |
\begin{isabelle}% |
156 |
{\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharminus}\ n\ {\isacharasterisk}\ k% |
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157 |
\end{isabelle} |
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\rulename{diff_mult_distrib} |
10602 | 159 |
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11187 | 160 |
\begin{isabelle}% |
161 |
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} |
10602 | 164 |
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11187 | 165 |
\begin{isabelle}% |
166 |
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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10602 | 167 |
\end{isabelle} |
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\rulename{nat_diff_split}% |
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\end{isamarkuptext}% |
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17175 | 170 |
\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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\ {\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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% |
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17056 | 191 |
\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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199 |
\isanewline |
17175 | 200 |
\isacommand{lemma}\isamarkupfalse% |
201 |
\ {\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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17056 | 202 |
% |
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\isadelimproof |
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% |
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\endisadelimproof |
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% |
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207 |
\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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16353 | 210 |
\ % |
211 |
\isamarkupcmt{\begin{isabelle}% |
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\ {\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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213 |
\end{isabelle}% |
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} |
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215 |
\isanewline |
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17175 | 216 |
\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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\isacommand{done}\isamarkupfalse% |
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219 |
% |
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17056 | 220 |
\endisatagproof |
221 |
{\isafoldproof}% |
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222 |
% |
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223 |
\isadelimproof |
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% |
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225 |
\endisadelimproof |
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11866 | 226 |
% |
11187 | 227 |
\begin{isamarkuptext}% |
228 |
\begin{isabelle}% |
|
229 |
m\ mod\ n\ {\isacharequal}\ {\isacharparenleft}if\ m\ {\isacharless}\ n\ then\ m\ else\ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ mod\ n{\isacharparenright}% |
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230 |
\end{isabelle} |
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11174 | 231 |
\rulename{mod_if} |
11187 | 232 |
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233 |
\begin{isabelle}% |
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27658 | 234 |
a\ div\ b\ {\isacharasterisk}\ b\ {\isacharplus}\ a\ mod\ b\ {\isacharequal}\ a% |
10602 | 235 |
\end{isabelle} |
11174 | 236 |
\rulename{mod_div_equality} |
10602 | 237 |
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238 |
||
11187 | 239 |
\begin{isabelle}% |
240 |
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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241 |
\end{isabelle} |
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11174 | 242 |
\rulename{div_mult1_eq} |
10602 | 243 |
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11187 | 244 |
\begin{isabelle}% |
245 |
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
|
246 |
\end{isabelle} |
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30224 | 247 |
\rulename{mod_mult_right_eq} |
10602 | 248 |
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11187 | 249 |
\begin{isabelle}% |
250 |
a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
|
251 |
\end{isabelle} |
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11174 | 252 |
\rulename{div_mult2_eq} |
10602 | 253 |
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11187 | 254 |
\begin{isabelle}% |
255 |
a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b% |
|
256 |
\end{isabelle} |
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11174 | 257 |
\rulename{mod_mult2_eq} |
10602 | 258 |
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11187 | 259 |
\begin{isabelle}% |
260 |
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ c\ {\isacharasterisk}\ a\ div\ {\isacharparenleft}c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ a\ div\ b% |
|
261 |
\end{isabelle} |
|
11174 | 262 |
\rulename{div_mult_mult1} |
10602 | 263 |
|
11187 | 264 |
\begin{isabelle}% |
27658 | 265 |
a\ div\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}% |
11187 | 266 |
\end{isabelle} |
27658 | 267 |
\rulename{div_by_0} |
10602 | 268 |
|
11187 | 269 |
\begin{isabelle}% |
27658 | 270 |
a\ mod\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharequal}\ a% |
11187 | 271 |
\end{isabelle} |
27658 | 272 |
\rulename{mod_by_0} |
10602 | 273 |
|
11187 | 274 |
\begin{isabelle}% |
275 |
{\isasymlbrakk}m\ dvd\ n{\isacharsemicolon}\ n\ dvd\ m{\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n% |
|
276 |
\end{isabelle} |
|
11174 | 277 |
\rulename{dvd_anti_sym} |
10602 | 278 |
|
11187 | 279 |
\begin{isabelle}% |
27658 | 280 |
{\isasymlbrakk}a\ dvd\ b{\isacharsemicolon}\ a\ dvd\ c{\isasymrbrakk}\ {\isasymLongrightarrow}\ a\ dvd\ b\ {\isacharplus}\ c% |
11187 | 281 |
\end{isabelle} |
11174 | 282 |
\rulename{dvd_add} |
10602 | 283 |
|
284 |
For the integers, I'd list a few theorems that somehow involve negative |
|
13758 | 285 |
numbers.% |
286 |
\end{isamarkuptext}% |
|
287 |
\isamarkuptrue% |
|
288 |
% |
|
289 |
\begin{isamarkuptext}% |
|
10602 | 290 |
Division, remainder of negatives |
291 |
||
292 |
||
11187 | 293 |
\begin{isabelle}% |
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294 |
{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ a\ mod\ b% |
11187 | 295 |
\end{isabelle} |
11174 | 296 |
\rulename{pos_mod_sign} |
10602 | 297 |
|
11187 | 298 |
\begin{isabelle}% |
11870
181bd2050cf4
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|
299 |
{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b% |
11174 | 300 |
\end{isabelle} |
11187 | 301 |
\rulename{pos_mod_bound} |
10602 | 302 |
|
11187 | 303 |
\begin{isabelle}% |
11870
181bd2050cf4
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paulson
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11866
diff
changeset
|
304 |
b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ {\isadigit{0}}% |
11174 | 305 |
\end{isabelle} |
11187 | 306 |
\rulename{neg_mod_sign} |
10602 | 307 |
|
11187 | 308 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
309 |
b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ b\ {\isacharless}\ a\ mod\ b% |
11187 | 310 |
\end{isabelle} |
11174 | 311 |
\rulename{neg_mod_bound} |
10602 | 312 |
|
11187 | 313 |
\begin{isabelle}% |
314 |
{\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 | 315 |
\end{isabelle} |
11187 | 316 |
\rulename{zdiv_zadd1_eq} |
10602 | 317 |
|
11187 | 318 |
\begin{isabelle}% |
319 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ mod\ c\ {\isacharequal}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ mod\ c% |
|
320 |
\end{isabelle} |
|
30224 | 321 |
\rulename{mod_add_eq} |
10602 | 322 |
|
11187 | 323 |
\begin{isabelle}% |
324 |
a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c% |
|
10602 | 325 |
\end{isabelle} |
11187 | 326 |
\rulename{zdiv_zmult1_eq} |
10602 | 327 |
|
11187 | 328 |
\begin{isabelle}% |
329 |
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
|
330 |
\end{isabelle} |
|
10602 | 331 |
\rulename{zmod_zmult1_eq} |
332 |
||
11187 | 333 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
334 |
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
11174 | 335 |
\end{isabelle} |
11187 | 336 |
\rulename{zdiv_zmult2_eq} |
10602 | 337 |
|
11187 | 338 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
diff
changeset
|
339 |
{\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 | 340 |
\end{isabelle} |
14400 | 341 |
\rulename{zmod_zmult2_eq}% |
11187 | 342 |
\end{isamarkuptext}% |
17175 | 343 |
\isamarkuptrue% |
344 |
\isacommand{lemma}\isamarkupfalse% |
|
345 |
\ {\isachardoublequoteopen}abs\ {\isacharparenleft}x{\isacharplus}y{\isacharparenright}\ {\isasymle}\ abs\ x\ {\isacharplus}\ abs\ {\isacharparenleft}y\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
17056 | 346 |
% |
347 |
\isadelimproof |
|
348 |
% |
|
349 |
\endisadelimproof |
|
350 |
% |
|
351 |
\isatagproof |
|
17175 | 352 |
\isacommand{by}\isamarkupfalse% |
353 |
\ arith% |
|
17056 | 354 |
\endisatagproof |
355 |
{\isafoldproof}% |
|
356 |
% |
|
357 |
\isadelimproof |
|
358 |
\isanewline |
|
359 |
% |
|
360 |
\endisadelimproof |
|
15481 | 361 |
\isanewline |
17175 | 362 |
\isacommand{lemma}\isamarkupfalse% |
363 |
\ {\isachardoublequoteopen}abs\ {\isacharparenleft}{\isadigit{2}}{\isacharasterisk}x{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ abs\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
17056 | 364 |
% |
365 |
\isadelimproof |
|
366 |
% |
|
367 |
\endisadelimproof |
|
368 |
% |
|
369 |
\isatagproof |
|
17175 | 370 |
\isacommand{by}\isamarkupfalse% |
371 |
\ {\isacharparenleft}simp\ add{\isacharcolon}\ abs{\isacharunderscore}if{\isacharparenright}% |
|
17056 | 372 |
\endisatagproof |
373 |
{\isafoldproof}% |
|
374 |
% |
|
375 |
\isadelimproof |
|
376 |
% |
|
377 |
\endisadelimproof |
|
11866 | 378 |
% |
11187 | 379 |
\begin{isamarkuptext}% |
13758 | 380 |
Induction rules for the Integers |
381 |
||
382 |
\begin{isabelle}% |
|
383 |
{\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% |
|
384 |
\end{isabelle} |
|
385 |
\rulename{int_ge_induct} |
|
386 |
||
387 |
\begin{isabelle}% |
|
388 |
{\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% |
|
389 |
\end{isabelle} |
|
390 |
\rulename{int_gr_induct} |
|
391 |
||
392 |
\begin{isabelle}% |
|
393 |
{\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% |
|
394 |
\end{isabelle} |
|
395 |
\rulename{int_le_induct} |
|
396 |
||
397 |
\begin{isabelle}% |
|
398 |
{\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% |
|
399 |
\end{isabelle} |
|
400 |
\rulename{int_less_induct}% |
|
401 |
\end{isamarkuptext}% |
|
402 |
\isamarkuptrue% |
|
403 |
% |
|
404 |
\begin{isamarkuptext}% |
|
14400 | 405 |
FIELDS |
10776 | 406 |
|
11187 | 407 |
\begin{isabelle}% |
25056 | 408 |
x\ {\isacharless}\ y\ {\isasymLongrightarrow}\ {\isasymexists}z{\isachargreater}x{\isachardot}\ z\ {\isacharless}\ y% |
11187 | 409 |
\end{isabelle} |
14295 | 410 |
\rulename{dense} |
10776 | 411 |
|
11187 | 412 |
\begin{isabelle}% |
14288 | 413 |
a\ {\isacharasterisk}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ b\ {\isacharslash}\ c% |
11187 | 414 |
\end{isabelle} |
14288 | 415 |
\rulename{times_divide_eq_right} |
10776 | 416 |
|
11187 | 417 |
\begin{isabelle}% |
14288 | 418 |
b\ {\isacharslash}\ c\ {\isacharasterisk}\ a\ {\isacharequal}\ b\ {\isacharasterisk}\ a\ {\isacharslash}\ c% |
11187 | 419 |
\end{isabelle} |
14288 | 420 |
\rulename{times_divide_eq_left} |
10776 | 421 |
|
11187 | 422 |
\begin{isabelle}% |
14288 | 423 |
a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ c\ {\isacharslash}\ b% |
11187 | 424 |
\end{isabelle} |
14288 | 425 |
\rulename{divide_divide_eq_right} |
10776 | 426 |
|
11187 | 427 |
\begin{isabelle}% |
14288 | 428 |
a\ {\isacharslash}\ b\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}% |
11187 | 429 |
\end{isabelle} |
14288 | 430 |
\rulename{divide_divide_eq_left} |
10776 | 431 |
|
11187 | 432 |
\begin{isabelle}% |
14295 | 433 |
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharminus}\ a\ {\isacharslash}\ b% |
11187 | 434 |
\end{isabelle} |
14295 | 435 |
\rulename{minus_divide_left} |
10776 | 436 |
|
11187 | 437 |
\begin{isabelle}% |
14295 | 438 |
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharminus}\ b% |
11187 | 439 |
\end{isabelle} |
14295 | 440 |
\rulename{minus_divide_right} |
10776 | 441 |
|
442 |
This last NOT a simprule |
|
443 |
||
11187 | 444 |
\begin{isabelle}% |
14295 | 445 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ c\ {\isacharplus}\ b\ {\isacharslash}\ c% |
11174 | 446 |
\end{isabelle} |
14295 | 447 |
\rulename{add_divide_distrib}% |
11187 | 448 |
\end{isamarkuptext}% |
17175 | 449 |
\isamarkuptrue% |
450 |
\isacommand{lemma}\isamarkupfalse% |
|
451 |
\ {\isachardoublequoteopen}{\isadigit{3}}{\isacharslash}{\isadigit{4}}\ {\isacharless}\ {\isacharparenleft}{\isadigit{7}}{\isacharslash}{\isadigit{8}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequoteclose}\isanewline |
|
17056 | 452 |
% |
453 |
\isadelimproof |
|
454 |
% |
|
455 |
\endisadelimproof |
|
456 |
% |
|
457 |
\isatagproof |
|
17175 | 458 |
\isacommand{by}\isamarkupfalse% |
459 |
\ simp% |
|
17056 | 460 |
\endisatagproof |
461 |
{\isafoldproof}% |
|
462 |
% |
|
463 |
\isadelimproof |
|
464 |
\ \isanewline |
|
465 |
% |
|
466 |
\endisadelimproof |
|
11174 | 467 |
\isanewline |
17175 | 468 |
\isacommand{lemma}\isamarkupfalse% |
469 |
\ {\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 | 470 |
\isadelimproof |
471 |
% |
|
472 |
\endisadelimproof |
|
473 |
% |
|
474 |
\isatagproof |
|
16353 | 475 |
% |
476 |
\begin{isamarkuptxt}% |
|
477 |
\begin{isabelle}% |
|
478 |
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\isacharparenright}% |
|
479 |
\end{isabelle}% |
|
480 |
\end{isamarkuptxt}% |
|
17175 | 481 |
\isamarkuptrue% |
482 |
\isacommand{apply}\isamarkupfalse% |
|
483 |
\ simp% |
|
16353 | 484 |
\begin{isamarkuptxt}% |
485 |
\begin{isabelle}% |
|
486 |
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{2}}\ {\isacharslash}\ {\isadigit{5}}{\isacharparenright}% |
|
487 |
\end{isabelle}% |
|
488 |
\end{isamarkuptxt}% |
|
17175 | 489 |
\isamarkuptrue% |
490 |
\isacommand{oops}\isamarkupfalse% |
|
491 |
% |
|
17056 | 492 |
\endisatagproof |
493 |
{\isafoldproof}% |
|
494 |
% |
|
495 |
\isadelimproof |
|
496 |
% |
|
497 |
\endisadelimproof |
|
498 |
\isanewline |
|
15481 | 499 |
\isanewline |
17175 | 500 |
\isacommand{lemma}\isamarkupfalse% |
501 |
\ {\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 | 502 |
\isadelimproof |
503 |
% |
|
504 |
\endisadelimproof |
|
505 |
% |
|
506 |
\isatagproof |
|
16353 | 507 |
% |
508 |
\begin{isamarkuptxt}% |
|
509 |
\begin{isabelle}% |
|
510 |
\ {\isadigit{1}}{\isachardot}\ {\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ x% |
|
511 |
\end{isabelle}% |
|
512 |
\end{isamarkuptxt}% |
|
17175 | 513 |
\isamarkuptrue% |
514 |
\isacommand{apply}\isamarkupfalse% |
|
515 |
\ simp% |
|
16353 | 516 |
\begin{isamarkuptxt}% |
517 |
\begin{isabelle}% |
|
518 |
\ {\isadigit{1}}{\isachardot}\ {\isadigit{2}}\ {\isacharless}\ x\ {\isacharasterisk}\ {\isadigit{5}}% |
|
519 |
\end{isabelle}% |
|
520 |
\end{isamarkuptxt}% |
|
17175 | 521 |
\isamarkuptrue% |
522 |
\isacommand{oops}\isamarkupfalse% |
|
523 |
% |
|
17056 | 524 |
\endisatagproof |
525 |
{\isafoldproof}% |
|
526 |
% |
|
527 |
\isadelimproof |
|
528 |
% |
|
529 |
\endisadelimproof |
|
14400 | 530 |
% |
531 |
\begin{isamarkuptext}% |
|
532 |
Ring and Field |
|
533 |
||
534 |
Requires a field, or else an ordered ring |
|
535 |
||
536 |
\begin{isabelle}% |
|
537 |
{\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}% |
|
538 |
\end{isabelle} |
|
539 |
\rulename{mult_eq_0_iff} |
|
540 |
||
541 |
\begin{isabelle}% |
|
542 |
{\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}% |
|
543 |
\end{isabelle} |
|
544 |
\rulename{mult_cancel_right} |
|
545 |
||
546 |
\begin{isabelle}% |
|
23504 | 547 |
{\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 | 548 |
\end{isabelle} |
23504 | 549 |
\rulename{mult_cancel_left}% |
14400 | 550 |
\end{isamarkuptext}% |
17175 | 551 |
\isamarkuptrue% |
17056 | 552 |
% |
553 |
\isadelimML |
|
554 |
% |
|
555 |
\endisadelimML |
|
556 |
% |
|
557 |
\isatagML |
|
17175 | 558 |
\isacommand{ML}\isamarkupfalse% |
17181 | 559 |
{\isacharverbatimopen}set\ show{\isacharunderscore}sorts{\isacharverbatimclose}% |
17056 | 560 |
\endisatagML |
561 |
{\isafoldML}% |
|
562 |
% |
|
563 |
\isadelimML |
|
564 |
% |
|
565 |
\endisadelimML |
|
14400 | 566 |
% |
567 |
\begin{isamarkuptext}% |
|
568 |
effect of show sorts on the above |
|
569 |
||
570 |
\begin{isabelle}% |
|
28838
d5db6dfcb34a
moved table of standard Isabelle symbols to isar-ref manual;
wenzelm
parents:
27658
diff
changeset
|
571 |
{\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 | 572 |
\end{isabelle} |
23504 | 573 |
\rulename{mult_cancel_left}% |
14400 | 574 |
\end{isamarkuptext}% |
17175 | 575 |
\isamarkuptrue% |
17056 | 576 |
% |
577 |
\isadelimML |
|
578 |
% |
|
579 |
\endisadelimML |
|
580 |
% |
|
581 |
\isatagML |
|
17175 | 582 |
\isacommand{ML}\isamarkupfalse% |
17181 | 583 |
{\isacharverbatimopen}reset\ show{\isacharunderscore}sorts{\isacharverbatimclose}% |
17056 | 584 |
\endisatagML |
585 |
{\isafoldML}% |
|
586 |
% |
|
587 |
\isadelimML |
|
588 |
% |
|
589 |
\endisadelimML |
|
14400 | 590 |
% |
591 |
\begin{isamarkuptext}% |
|
592 |
absolute value |
|
593 |
||
594 |
\begin{isabelle}% |
|
595 |
{\isasymbar}a\ {\isacharasterisk}\ b{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}b{\isasymbar}% |
|
596 |
\end{isabelle} |
|
597 |
\rulename{abs_mult} |
|
598 |
||
599 |
\begin{isabelle}% |
|
600 |
{\isacharparenleft}{\isasymbar}a{\isasymbar}\ {\isasymle}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isasymle}\ b\ {\isasymand}\ {\isacharminus}\ a\ {\isasymle}\ b{\isacharparenright}% |
|
601 |
\end{isabelle} |
|
602 |
\rulename{abs_le_iff} |
|
603 |
||
604 |
\begin{isabelle}% |
|
605 |
{\isasymbar}a\ {\isacharplus}\ b{\isasymbar}\ {\isasymle}\ {\isasymbar}a{\isasymbar}\ {\isacharplus}\ {\isasymbar}b{\isasymbar}% |
|
606 |
\end{isabelle} |
|
607 |
\rulename{abs_triangle_ineq} |
|
608 |
||
609 |
\begin{isabelle}% |
|
610 |
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharequal}\ a\ {\isacharcircum}\ m\ {\isacharasterisk}\ a\ {\isacharcircum}\ n% |
|
611 |
\end{isabelle} |
|
612 |
\rulename{power_add} |
|
613 |
||
614 |
\begin{isabelle}% |
|
615 |
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isacharcircum}\ m{\isacharparenright}\ {\isacharcircum}\ n% |
|
616 |
\end{isabelle} |
|
617 |
\rulename{power_mult} |
|
618 |
||
619 |
\begin{isabelle}% |
|
620 |
{\isasymbar}a\ {\isacharcircum}\ n{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharcircum}\ n% |
|
621 |
\end{isabelle} |
|
622 |
\rulename{power_abs}% |
|
623 |
\end{isamarkuptext}% |
|
17175 | 624 |
\isamarkuptrue% |
17056 | 625 |
% |
626 |
\isadelimtheory |
|
627 |
% |
|
628 |
\endisadelimtheory |
|
629 |
% |
|
630 |
\isatagtheory |
|
17175 | 631 |
\isacommand{end}\isamarkupfalse% |
632 |
% |
|
17056 | 633 |
\endisatagtheory |
634 |
{\isafoldtheory}% |
|
635 |
% |
|
636 |
\isadelimtheory |
|
637 |
% |
|
638 |
\endisadelimtheory |
|
639 |
\isanewline |
|
11187 | 640 |
\end{isabellebody}% |
10602 | 641 |
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642 |
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643 |
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|
644 |
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