author | paulson |
Tue, 28 Jun 2005 15:26:32 +0200 | |
changeset 16585 | 02cf78f0afce |
parent 16523 | f8a734dc0fbc |
child 17056 | 05fc32a23b8b |
permissions | -rw-r--r-- |
10602 | 1 |
% |
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\begin{isabellebody}% |
3 |
\def\isabellecontext{Numbers}% |
|
10602 | 4 |
\isanewline |
16523 | 5 |
\isacommand{theory}\ Numbers\ \isakeyword{imports}\ Real\ \isakeyword{begin}\isanewline |
10602 | 6 |
\isanewline |
11866 | 7 |
\isamarkupfalse% |
16353 | 8 |
\isacommand{ML}\ {\isachardoublequote}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequote}\isanewline |
11866 | 9 |
\isamarkupfalse% |
16353 | 10 |
\isacommand{ML}\ {\isachardoublequote}IsarOutput{\isachardot}indent\ {\isacharcolon}{\isacharequal}\ {\isadigit{0}}{\isachardoublequote}\isamarkupfalse% |
11866 | 11 |
% |
11187 | 12 |
\begin{isamarkuptext}% |
10602 | 13 |
numeric literals; default simprules; can re-orient% |
11187 | 14 |
\end{isamarkuptext}% |
11866 | 15 |
\isamarkuptrue% |
16 |
\isacommand{lemma}\ {\isachardoublequote}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequote}\isamarkupfalse% |
|
16353 | 17 |
% |
18 |
\begin{isamarkuptxt}% |
|
19 |
\begin{isabelle}% |
|
20 |
\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isadigit{2}}{\isasymColon}{\isacharprime}a{\isacharparenright}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m% |
|
21 |
\end{isabelle}% |
|
22 |
\end{isamarkuptxt}% |
|
11866 | 23 |
\isamarkuptrue% |
16353 | 24 |
\isacommand{oops}\isanewline |
15481 | 25 |
\isanewline |
11866 | 26 |
\isamarkupfalse% |
11187 | 27 |
\isacommand{consts}\ h\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline |
11866 | 28 |
\isamarkupfalse% |
11187 | 29 |
\isacommand{recdef}\ h\ {\isachardoublequote}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequote}\isanewline |
11866 | 30 |
{\isachardoublequote}h\ i\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isacharequal}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequote}\isamarkupfalse% |
31 |
% |
|
11187 | 32 |
\begin{isamarkuptext}% |
11708 | 33 |
\isa{h\ {\isadigit{3}}\ {\isacharequal}\ {\isadigit{2}}} |
11187 | 34 |
\isa{h\ i\ {\isacharequal}\ i}% |
35 |
\end{isamarkuptext}% |
|
11866 | 36 |
\isamarkuptrue% |
10878 | 37 |
% |
11187 | 38 |
\begin{isamarkuptext}% |
39 |
\begin{isabelle}% |
|
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Numeral{\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}% |
11187 | 41 |
\end{isabelle} |
11174 | 42 |
\rulename{numeral_0_eq_0} |
10602 | 43 |
|
11187 | 44 |
\begin{isabelle}% |
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45 |
Numeral{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}{\isasymColon}{\isacharprime}a{\isacharparenright}% |
10602 | 46 |
\end{isabelle} |
11187 | 47 |
\rulename{numeral_1_eq_1} |
11174 | 48 |
|
11187 | 49 |
\begin{isabelle}% |
11708 | 50 |
{\isadigit{2}}\ {\isacharplus}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}% |
11187 | 51 |
\end{isabelle} |
10602 | 52 |
\rulename{add_2_eq_Suc} |
11187 | 53 |
|
54 |
\begin{isabelle}% |
|
11708 | 55 |
n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}% |
10602 | 56 |
\end{isabelle} |
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\rulename{add_2_eq_Suc'} |
58 |
||
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\begin{isabelle}% |
14270 | 60 |
a\ {\isacharplus}\ b\ {\isacharplus}\ c\ {\isacharequal}\ a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}% |
11174 | 61 |
\end{isabelle} |
11187 | 62 |
\rulename{add_assoc} |
10602 | 63 |
|
11187 | 64 |
\begin{isabelle}% |
14270 | 65 |
a\ {\isacharplus}\ b\ {\isacharequal}\ b\ {\isacharplus}\ a% |
11174 | 66 |
\end{isabelle} |
11187 | 67 |
\rulename{add_commute} |
10602 | 68 |
|
11187 | 69 |
\begin{isabelle}% |
14270 | 70 |
a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharplus}\ {\isacharparenleft}a\ {\isacharplus}\ c{\isacharparenright}% |
11187 | 71 |
\end{isabelle} |
11174 | 72 |
\rulename{add_left_commute} |
10602 | 73 |
|
74 |
these form add_ac; similarly there is mult_ac% |
|
11187 | 75 |
\end{isamarkuptext}% |
11866 | 76 |
\isamarkuptrue% |
77 |
\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}\isamarkupfalse% |
|
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% |
79 |
\begin{isamarkuptxt}% |
|
80 |
\begin{isabelle}% |
|
81 |
\ {\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}% |
|
82 |
\end{isabelle}% |
|
83 |
\end{isamarkuptxt}% |
|
11866 | 84 |
\isamarkuptrue% |
16353 | 85 |
\isacommand{apply}\ {\isacharparenleft}simp\ add{\isacharcolon}\ add{\isacharunderscore}ac\ mult{\isacharunderscore}ac{\isacharparenright}\isamarkupfalse% |
86 |
% |
|
87 |
\begin{isamarkuptxt}% |
|
88 |
\begin{isabelle}% |
|
89 |
\ {\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}% |
|
91 |
\end{isabelle}% |
|
92 |
\end{isamarkuptxt}% |
|
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\isamarkuptrue% |
16353 | 94 |
\isacommand{oops}\isamarkupfalse% |
11866 | 95 |
% |
11187 | 96 |
\begin{isamarkuptext}% |
97 |
\begin{isabelle}% |
|
98 |
m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\isasymle}\ n\ div\ k% |
|
99 |
\end{isabelle} |
|
11174 | 100 |
\rulename{div_le_mono} |
10602 | 101 |
|
11187 | 102 |
\begin{isabelle}% |
103 |
{\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharminus}\ n\ {\isacharasterisk}\ k% |
|
104 |
\end{isabelle} |
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11174 | 105 |
\rulename{diff_mult_distrib} |
10602 | 106 |
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11187 | 107 |
\begin{isabelle}% |
108 |
m\ mod\ n\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ mod\ {\isacharparenleft}n\ {\isacharasterisk}\ k{\isacharparenright}% |
|
109 |
\end{isabelle} |
|
11174 | 110 |
\rulename{mod_mult_distrib} |
10602 | 111 |
|
11187 | 112 |
\begin{isabelle}% |
113 |
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}% |
|
10602 | 114 |
\end{isabelle} |
11187 | 115 |
\rulename{nat_diff_split}% |
116 |
\end{isamarkuptext}% |
|
11866 | 117 |
\isamarkuptrue% |
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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}\ {\isacharparenleft}{\isadigit{1}}{\isacharcolon}{\isacharcolon}nat{\isacharparenright}{\isachardoublequote}\isanewline |
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119 |
\isamarkupfalse% |
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\isacommand{apply}\ {\isacharparenleft}clarsimp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split\ iff\ del{\isacharcolon}\ less{\isacharunderscore}Suc{\isadigit{0}}{\isacharparenright}\isanewline |
121 |
\ % |
|
122 |
\isamarkupcmt{\begin{isabelle}% |
|
123 |
\ {\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}}% |
|
124 |
\end{isabelle}% |
|
125 |
} |
|
126 |
\isanewline |
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\isamarkupfalse% |
16353 | 128 |
\isacommand{apply}\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}n{\isacharequal}{\isadigit{0}}{\isachardoublequote}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline |
15481 | 129 |
\isamarkupfalse% |
16353 | 130 |
\isacommand{done}\isanewline |
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\isanewline |
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\isanewline |
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133 |
\isamarkupfalse% |
11708 | 134 |
\isacommand{lemma}\ {\isachardoublequote}{\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}{\isachardoublequote}\isanewline |
11866 | 135 |
\isamarkupfalse% |
16353 | 136 |
\isacommand{apply}\ {\isacharparenleft}simp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharcomma}\ clarify{\isacharparenright}\isanewline |
137 |
\ % |
|
138 |
\isamarkupcmt{\begin{isabelle}% |
|
139 |
\ {\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}}% |
|
140 |
\end{isabelle}% |
|
141 |
} |
|
142 |
\isanewline |
|
11866 | 143 |
\isamarkupfalse% |
16353 | 144 |
\isacommand{apply}\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}n{\isacharequal}{\isadigit{0}}\ {\isacharbar}\ n{\isacharequal}{\isadigit{1}}{\isachardoublequote}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline |
11866 | 145 |
\isamarkupfalse% |
16353 | 146 |
\isacommand{done}\isamarkupfalse% |
11866 | 147 |
% |
11187 | 148 |
\begin{isamarkuptext}% |
149 |
\begin{isabelle}% |
|
150 |
m\ mod\ n\ {\isacharequal}\ {\isacharparenleft}if\ m\ {\isacharless}\ n\ then\ m\ else\ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ mod\ n{\isacharparenright}% |
|
151 |
\end{isabelle} |
|
11174 | 152 |
\rulename{mod_if} |
11187 | 153 |
|
154 |
\begin{isabelle}% |
|
155 |
m\ div\ n\ {\isacharasterisk}\ n\ {\isacharplus}\ m\ mod\ n\ {\isacharequal}\ m% |
|
10602 | 156 |
\end{isabelle} |
11174 | 157 |
\rulename{mod_div_equality} |
10602 | 158 |
|
159 |
||
11187 | 160 |
\begin{isabelle}% |
161 |
a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c% |
|
162 |
\end{isabelle} |
|
11174 | 163 |
\rulename{div_mult1_eq} |
10602 | 164 |
|
11187 | 165 |
\begin{isabelle}% |
166 |
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
|
167 |
\end{isabelle} |
|
11174 | 168 |
\rulename{mod_mult1_eq} |
10602 | 169 |
|
11187 | 170 |
\begin{isabelle}% |
171 |
a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
|
172 |
\end{isabelle} |
|
11174 | 173 |
\rulename{div_mult2_eq} |
10602 | 174 |
|
11187 | 175 |
\begin{isabelle}% |
176 |
a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b% |
|
177 |
\end{isabelle} |
|
11174 | 178 |
\rulename{mod_mult2_eq} |
10602 | 179 |
|
11187 | 180 |
\begin{isabelle}% |
181 |
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ c\ {\isacharasterisk}\ a\ div\ {\isacharparenleft}c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ a\ div\ b% |
|
182 |
\end{isabelle} |
|
11174 | 183 |
\rulename{div_mult_mult1} |
10602 | 184 |
|
11187 | 185 |
\begin{isabelle}% |
186 |
a\ div\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}% |
|
187 |
\end{isabelle} |
|
11174 | 188 |
\rulename{DIVISION_BY_ZERO_DIV} |
10602 | 189 |
|
11187 | 190 |
\begin{isabelle}% |
191 |
a\ mod\ {\isadigit{0}}\ {\isacharequal}\ a% |
|
192 |
\end{isabelle} |
|
11174 | 193 |
\rulename{DIVISION_BY_ZERO_MOD} |
10602 | 194 |
|
11187 | 195 |
\begin{isabelle}% |
196 |
{\isasymlbrakk}m\ dvd\ n{\isacharsemicolon}\ n\ dvd\ m{\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n% |
|
197 |
\end{isabelle} |
|
11174 | 198 |
\rulename{dvd_anti_sym} |
10602 | 199 |
|
11187 | 200 |
\begin{isabelle}% |
201 |
{\isasymlbrakk}k\ dvd\ m{\isacharsemicolon}\ k\ dvd\ n{\isasymrbrakk}\ {\isasymLongrightarrow}\ k\ dvd\ m\ {\isacharplus}\ n% |
|
202 |
\end{isabelle} |
|
11174 | 203 |
\rulename{dvd_add} |
10602 | 204 |
|
205 |
For the integers, I'd list a few theorems that somehow involve negative |
|
13758 | 206 |
numbers.% |
207 |
\end{isamarkuptext}% |
|
208 |
\isamarkuptrue% |
|
209 |
% |
|
210 |
\begin{isamarkuptext}% |
|
10602 | 211 |
Division, remainder of negatives |
212 |
||
213 |
||
11187 | 214 |
\begin{isabelle}% |
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215 |
{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ a\ mod\ b% |
11187 | 216 |
\end{isabelle} |
11174 | 217 |
\rulename{pos_mod_sign} |
10602 | 218 |
|
11187 | 219 |
\begin{isabelle}% |
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220 |
{\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b% |
11174 | 221 |
\end{isabelle} |
11187 | 222 |
\rulename{pos_mod_bound} |
10602 | 223 |
|
11187 | 224 |
\begin{isabelle}% |
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225 |
b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ {\isadigit{0}}% |
11174 | 226 |
\end{isabelle} |
11187 | 227 |
\rulename{neg_mod_sign} |
10602 | 228 |
|
11187 | 229 |
\begin{isabelle}% |
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230 |
b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ b\ {\isacharless}\ a\ mod\ b% |
11187 | 231 |
\end{isabelle} |
11174 | 232 |
\rulename{neg_mod_bound} |
10602 | 233 |
|
11187 | 234 |
\begin{isabelle}% |
235 |
{\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 | 236 |
\end{isabelle} |
11187 | 237 |
\rulename{zdiv_zadd1_eq} |
10602 | 238 |
|
11187 | 239 |
\begin{isabelle}% |
240 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ mod\ c\ {\isacharequal}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ mod\ c% |
|
241 |
\end{isabelle} |
|
10602 | 242 |
\rulename{zmod_zadd1_eq} |
243 |
||
11187 | 244 |
\begin{isabelle}% |
245 |
a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c% |
|
10602 | 246 |
\end{isabelle} |
11187 | 247 |
\rulename{zdiv_zmult1_eq} |
10602 | 248 |
|
11187 | 249 |
\begin{isabelle}% |
250 |
a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c% |
|
251 |
\end{isabelle} |
|
10602 | 252 |
\rulename{zmod_zmult1_eq} |
253 |
||
11187 | 254 |
\begin{isabelle}% |
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|
255 |
{\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c% |
11174 | 256 |
\end{isabelle} |
11187 | 257 |
\rulename{zdiv_zmult2_eq} |
10602 | 258 |
|
11187 | 259 |
\begin{isabelle}% |
11870
181bd2050cf4
Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents:
11866
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changeset
|
260 |
{\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 | 261 |
\end{isabelle} |
14400 | 262 |
\rulename{zmod_zmult2_eq}% |
11187 | 263 |
\end{isamarkuptext}% |
11866 | 264 |
\isamarkuptrue% |
11187 | 265 |
\isacommand{lemma}\ {\isachardoublequote}abs\ {\isacharparenleft}x{\isacharplus}y{\isacharparenright}\ {\isasymle}\ abs\ x\ {\isacharplus}\ abs\ {\isacharparenleft}y\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline |
11866 | 266 |
\isamarkupfalse% |
16353 | 267 |
\isacommand{by}\ arith\isanewline |
15481 | 268 |
\isanewline |
11866 | 269 |
\isamarkupfalse% |
11708 | 270 |
\isacommand{lemma}\ {\isachardoublequote}abs\ {\isacharparenleft}{\isadigit{2}}{\isacharasterisk}x{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ abs\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline |
11866 | 271 |
\isamarkupfalse% |
16585 | 272 |
\isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ abs{\isacharunderscore}if{\isacharparenright}\isamarkupfalse% |
11866 | 273 |
% |
11187 | 274 |
\begin{isamarkuptext}% |
13758 | 275 |
Induction rules for the Integers |
276 |
||
277 |
\begin{isabelle}% |
|
278 |
{\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% |
|
279 |
\end{isabelle} |
|
280 |
\rulename{int_ge_induct} |
|
281 |
||
282 |
\begin{isabelle}% |
|
283 |
{\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% |
|
284 |
\end{isabelle} |
|
285 |
\rulename{int_gr_induct} |
|
286 |
||
287 |
\begin{isabelle}% |
|
288 |
{\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% |
|
289 |
\end{isabelle} |
|
290 |
\rulename{int_le_induct} |
|
291 |
||
292 |
\begin{isabelle}% |
|
293 |
{\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% |
|
294 |
\end{isabelle} |
|
295 |
\rulename{int_less_induct}% |
|
296 |
\end{isamarkuptext}% |
|
297 |
\isamarkuptrue% |
|
298 |
% |
|
299 |
\begin{isamarkuptext}% |
|
14400 | 300 |
FIELDS |
10776 | 301 |
|
11187 | 302 |
\begin{isabelle}% |
15364 | 303 |
a\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isasymexists}r{\isachargreater}a{\isachardot}\ r\ {\isacharless}\ b% |
11187 | 304 |
\end{isabelle} |
14295 | 305 |
\rulename{dense} |
10776 | 306 |
|
11187 | 307 |
\begin{isabelle}% |
14288 | 308 |
a\ {\isacharasterisk}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ b\ {\isacharslash}\ c% |
11187 | 309 |
\end{isabelle} |
14288 | 310 |
\rulename{times_divide_eq_right} |
10776 | 311 |
|
11187 | 312 |
\begin{isabelle}% |
14288 | 313 |
b\ {\isacharslash}\ c\ {\isacharasterisk}\ a\ {\isacharequal}\ b\ {\isacharasterisk}\ a\ {\isacharslash}\ c% |
11187 | 314 |
\end{isabelle} |
14288 | 315 |
\rulename{times_divide_eq_left} |
10776 | 316 |
|
11187 | 317 |
\begin{isabelle}% |
14288 | 318 |
a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ c\ {\isacharslash}\ b% |
11187 | 319 |
\end{isabelle} |
14288 | 320 |
\rulename{divide_divide_eq_right} |
10776 | 321 |
|
11187 | 322 |
\begin{isabelle}% |
14288 | 323 |
a\ {\isacharslash}\ b\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}% |
11187 | 324 |
\end{isabelle} |
14288 | 325 |
\rulename{divide_divide_eq_left} |
10776 | 326 |
|
11187 | 327 |
\begin{isabelle}% |
14295 | 328 |
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharminus}\ a\ {\isacharslash}\ b% |
11187 | 329 |
\end{isabelle} |
14295 | 330 |
\rulename{minus_divide_left} |
10776 | 331 |
|
11187 | 332 |
\begin{isabelle}% |
14295 | 333 |
{\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharminus}\ b% |
11187 | 334 |
\end{isabelle} |
14295 | 335 |
\rulename{minus_divide_right} |
10776 | 336 |
|
337 |
This last NOT a simprule |
|
338 |
||
11187 | 339 |
\begin{isabelle}% |
14295 | 340 |
{\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ c\ {\isacharplus}\ b\ {\isacharslash}\ c% |
11174 | 341 |
\end{isabelle} |
14295 | 342 |
\rulename{add_divide_distrib}% |
11187 | 343 |
\end{isamarkuptext}% |
11866 | 344 |
\isamarkuptrue% |
11708 | 345 |
\isacommand{lemma}\ {\isachardoublequote}{\isadigit{3}}{\isacharslash}{\isadigit{4}}\ {\isacharless}\ {\isacharparenleft}{\isadigit{7}}{\isacharslash}{\isadigit{8}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}\isanewline |
11866 | 346 |
\isamarkupfalse% |
16353 | 347 |
\isacommand{by}\ simp\ \isanewline |
11174 | 348 |
\isanewline |
11866 | 349 |
\isamarkupfalse% |
350 |
\isacommand{lemma}\ {\isachardoublequote}P\ {\isacharparenleft}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isacharparenright}{\isachardoublequote}\isamarkupfalse% |
|
16353 | 351 |
% |
352 |
\begin{isamarkuptxt}% |
|
353 |
\begin{isabelle}% |
|
354 |
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\isacharparenright}% |
|
355 |
\end{isabelle}% |
|
356 |
\end{isamarkuptxt}% |
|
11866 | 357 |
\isamarkuptrue% |
16353 | 358 |
\isacommand{apply}\ simp\isamarkupfalse% |
359 |
% |
|
360 |
\begin{isamarkuptxt}% |
|
361 |
\begin{isabelle}% |
|
362 |
\ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{2}}\ {\isacharslash}\ {\isadigit{5}}{\isacharparenright}% |
|
363 |
\end{isabelle}% |
|
364 |
\end{isamarkuptxt}% |
|
11866 | 365 |
\isamarkuptrue% |
16353 | 366 |
\isacommand{oops}\isanewline |
15481 | 367 |
\isanewline |
11866 | 368 |
\isamarkupfalse% |
369 |
\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}{\isacharslash}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}\isamarkupfalse% |
|
16353 | 370 |
% |
371 |
\begin{isamarkuptxt}% |
|
372 |
\begin{isabelle}% |
|
373 |
\ {\isadigit{1}}{\isachardot}\ {\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ x% |
|
374 |
\end{isabelle}% |
|
375 |
\end{isamarkuptxt}% |
|
11866 | 376 |
\isamarkuptrue% |
16353 | 377 |
\isacommand{apply}\ simp\isamarkupfalse% |
378 |
% |
|
379 |
\begin{isamarkuptxt}% |
|
380 |
\begin{isabelle}% |
|
381 |
\ {\isadigit{1}}{\isachardot}\ {\isadigit{2}}\ {\isacharless}\ x\ {\isacharasterisk}\ {\isadigit{5}}% |
|
382 |
\end{isabelle}% |
|
383 |
\end{isamarkuptxt}% |
|
11866 | 384 |
\isamarkuptrue% |
16353 | 385 |
\isacommand{oops}\isamarkupfalse% |
14400 | 386 |
% |
387 |
\begin{isamarkuptext}% |
|
388 |
Ring and Field |
|
389 |
||
390 |
Requires a field, or else an ordered ring |
|
391 |
||
392 |
\begin{isabelle}% |
|
393 |
{\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}% |
|
394 |
\end{isabelle} |
|
395 |
\rulename{mult_eq_0_iff} |
|
396 |
||
397 |
\begin{isabelle}% |
|
398 |
{\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}% |
|
399 |
\end{isabelle} |
|
400 |
\rulename{field_mult_eq_0_iff} |
|
401 |
||
402 |
\begin{isabelle}% |
|
403 |
{\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}% |
|
404 |
\end{isabelle} |
|
405 |
\rulename{mult_cancel_right} |
|
406 |
||
407 |
\begin{isabelle}% |
|
408 |
{\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}% |
|
409 |
\end{isabelle} |
|
410 |
\rulename{field_mult_cancel_right}% |
|
411 |
\end{isamarkuptext}% |
|
412 |
\isamarkuptrue% |
|
16353 | 413 |
\isacommand{ML}{\isacharbraceleft}{\isacharasterisk}set\ show{\isacharunderscore}sorts{\isacharasterisk}{\isacharbraceright}\isamarkupfalse% |
14400 | 414 |
% |
415 |
\begin{isamarkuptext}% |
|
416 |
effect of show sorts on the above |
|
417 |
||
418 |
\begin{isabelle}% |
|
419 |
{\isacharparenleft}{\isacharparenleft}a{\isasymColon}{\isacharprime}a{\isasymColon}field{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}c{\isasymColon}{\isacharprime}a{\isasymColon}field{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}b{\isasymColon}{\isacharprime}a{\isasymColon}field{\isacharparenright}\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\isanewline |
|
420 |
{\isacharparenleft}c\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isasymColon}field{\isacharparenright}\ {\isasymor}\ a\ {\isacharequal}\ b{\isacharparenright}% |
|
421 |
\end{isabelle} |
|
422 |
\rulename{field_mult_cancel_right}% |
|
423 |
\end{isamarkuptext}% |
|
424 |
\isamarkuptrue% |
|
16353 | 425 |
\isacommand{ML}{\isacharbraceleft}{\isacharasterisk}reset\ show{\isacharunderscore}sorts{\isacharasterisk}{\isacharbraceright}\isamarkupfalse% |
14400 | 426 |
% |
427 |
\begin{isamarkuptext}% |
|
428 |
absolute value |
|
429 |
||
430 |
\begin{isabelle}% |
|
431 |
{\isasymbar}a\ {\isacharasterisk}\ b{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}b{\isasymbar}% |
|
432 |
\end{isabelle} |
|
433 |
\rulename{abs_mult} |
|
434 |
||
435 |
\begin{isabelle}% |
|
436 |
{\isacharparenleft}{\isasymbar}a{\isasymbar}\ {\isasymle}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isasymle}\ b\ {\isasymand}\ {\isacharminus}\ a\ {\isasymle}\ b{\isacharparenright}% |
|
437 |
\end{isabelle} |
|
438 |
\rulename{abs_le_iff} |
|
439 |
||
440 |
\begin{isabelle}% |
|
441 |
{\isasymbar}a\ {\isacharplus}\ b{\isasymbar}\ {\isasymle}\ {\isasymbar}a{\isasymbar}\ {\isacharplus}\ {\isasymbar}b{\isasymbar}% |
|
442 |
\end{isabelle} |
|
443 |
\rulename{abs_triangle_ineq} |
|
444 |
||
445 |
\begin{isabelle}% |
|
446 |
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharequal}\ a\ {\isacharcircum}\ m\ {\isacharasterisk}\ a\ {\isacharcircum}\ n% |
|
447 |
\end{isabelle} |
|
448 |
\rulename{power_add} |
|
449 |
||
450 |
\begin{isabelle}% |
|
451 |
a\ {\isacharcircum}\ {\isacharparenleft}m\ {\isacharasterisk}\ n{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}a\ {\isacharcircum}\ m{\isacharparenright}\ {\isacharcircum}\ n% |
|
452 |
\end{isabelle} |
|
453 |
\rulename{power_mult} |
|
454 |
||
455 |
\begin{isabelle}% |
|
456 |
{\isasymbar}a\ {\isacharcircum}\ n{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharcircum}\ n% |
|
457 |
\end{isabelle} |
|
458 |
\rulename{power_abs}% |
|
459 |
\end{isamarkuptext}% |
|
460 |
\isamarkuptrue% |
|
10602 | 461 |
\isacommand{end}\isanewline |
11866 | 462 |
\isamarkupfalse% |
11187 | 463 |
\end{isabellebody}% |
10602 | 464 |
%%% Local Variables: |
465 |
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466 |
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467 |
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