doc-src/TutorialI/Types/document/Numbers.tex
author paulson
Sun Feb 15 10:46:37 2004 +0100 (2004-02-15)
changeset 14387 e96d5c42c4b0
parent 14353 79f9fbef9106
child 14400 6069098854b9
permissions -rw-r--r--
Polymorphic treatment of binary arithmetic using axclasses
     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{Numbers}%
     4 \isanewline
     5 \isacommand{theory}\ Numbers\ {\isacharequal}\ Real{\isacharcolon}\isanewline
     6 \isanewline
     7 \isamarkupfalse%
     8 \isacommand{ML}\ {\isachardoublequote}Pretty{\isachardot}setmargin\ {\isadigit{6}}{\isadigit{4}}{\isachardoublequote}\isanewline
     9 \isamarkupfalse%
    10 \isacommand{ML}\ {\isachardoublequote}IsarOutput{\isachardot}indent\ {\isacharcolon}{\isacharequal}\ {\isadigit{0}}{\isachardoublequote}\isamarkupfalse%
    11 %
    12 \begin{isamarkuptext}%
    13 numeric literals; default simprules; can re-orient%
    14 \end{isamarkuptext}%
    15 \isamarkuptrue%
    16 \isacommand{lemma}\ {\isachardoublequote}{\isadigit{2}}\ {\isacharasterisk}\ m\ {\isacharequal}\ m\ {\isacharplus}\ m{\isachardoublequote}\isamarkupfalse%
    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}%
    23 \isamarkuptrue%
    24 \isacommand{oops}\isanewline
    25 \isanewline
    26 \isamarkupfalse%
    27 \isacommand{consts}\ h\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}nat\ {\isasymRightarrow}\ nat{\isachardoublequote}\isanewline
    28 \isamarkupfalse%
    29 \isacommand{recdef}\ h\ {\isachardoublequote}{\isacharbraceleft}{\isacharbraceright}{\isachardoublequote}\isanewline
    30 {\isachardoublequote}h\ i\ {\isacharequal}\ {\isacharparenleft}if\ i\ {\isacharequal}\ {\isadigit{3}}\ then\ {\isadigit{2}}\ else\ i{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
    31 %
    32 \begin{isamarkuptext}%
    33 \isa{h\ {\isadigit{3}}\ {\isacharequal}\ {\isadigit{2}}}
    34 \isa{h\ i\ {\isacharequal}\ i}%
    35 \end{isamarkuptext}%
    36 \isamarkuptrue%
    37 %
    38 \begin{isamarkuptext}%
    39 \begin{isabelle}%
    40 Numeral{\isadigit{0}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{0}}{\isasymColon}{\isacharprime}a{\isacharparenright}%
    41 \end{isabelle}
    42 \rulename{numeral_0_eq_0}
    43 
    44 \begin{isabelle}%
    45 Numeral{\isadigit{1}}\ {\isacharequal}\ {\isacharparenleft}{\isadigit{1}}{\isasymColon}{\isacharprime}a{\isacharparenright}%
    46 \end{isabelle}
    47 \rulename{numeral_1_eq_1}
    48 
    49 \begin{isabelle}%
    50 {\isadigit{2}}\ {\isacharplus}\ n\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}%
    51 \end{isabelle}
    52 \rulename{add_2_eq_Suc}
    53 
    54 \begin{isabelle}%
    55 n\ {\isacharplus}\ {\isadigit{2}}\ {\isacharequal}\ Suc\ {\isacharparenleft}Suc\ n{\isacharparenright}%
    56 \end{isabelle}
    57 \rulename{add_2_eq_Suc'}
    58 
    59 \begin{isabelle}%
    60 a\ {\isacharplus}\ b\ {\isacharplus}\ c\ {\isacharequal}\ a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}%
    61 \end{isabelle}
    62 \rulename{add_assoc}
    63 
    64 \begin{isabelle}%
    65 a\ {\isacharplus}\ b\ {\isacharequal}\ b\ {\isacharplus}\ a%
    66 \end{isabelle}
    67 \rulename{add_commute}
    68 
    69 \begin{isabelle}%
    70 a\ {\isacharplus}\ {\isacharparenleft}b\ {\isacharplus}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharplus}\ {\isacharparenleft}a\ {\isacharplus}\ c{\isacharparenright}%
    71 \end{isabelle}
    72 \rulename{add_left_commute}
    73 
    74 these form add_ac; similarly there is mult_ac%
    75 \end{isamarkuptext}%
    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%
    78 %
    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}%
    84 \isamarkuptrue%
    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
    90 \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}%
    93 \isamarkuptrue%
    94 \isacommand{oops}\isamarkupfalse%
    95 %
    96 \begin{isamarkuptext}%
    97 \begin{isabelle}%
    98 {\isasymlbrakk}i\ {\isasymle}\ j{\isacharsemicolon}\ k\ {\isasymle}\ l{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharasterisk}\ k\ {\isasymle}\ j\ {\isacharasterisk}\ l%
    99 \end{isabelle}
   100 \rulename{mult_le_mono}
   101 
   102 \begin{isabelle}%
   103 {\isasymlbrakk}i\ {\isacharless}\ j{\isacharsemicolon}\ {\isadigit{0}}\ {\isacharless}\ k{\isasymrbrakk}\ {\isasymLongrightarrow}\ i\ {\isacharasterisk}\ k\ {\isacharless}\ j\ {\isacharasterisk}\ k%
   104 \end{isabelle}
   105 \rulename{mult_less_mono1}
   106 
   107 \begin{isabelle}%
   108 m\ {\isasymle}\ n\ {\isasymLongrightarrow}\ m\ div\ k\ {\isasymle}\ n\ div\ k%
   109 \end{isabelle}
   110 \rulename{div_le_mono}
   111 
   112 \begin{isabelle}%
   113 {\isacharparenleft}m\ {\isacharplus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharplus}\ n\ {\isacharasterisk}\ k%
   114 \end{isabelle}
   115 \rulename{add_mult_distrib}
   116 
   117 \begin{isabelle}%
   118 {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ {\isacharminus}\ n\ {\isacharasterisk}\ k%
   119 \end{isabelle}
   120 \rulename{diff_mult_distrib}
   121 
   122 \begin{isabelle}%
   123 m\ mod\ n\ {\isacharasterisk}\ k\ {\isacharequal}\ m\ {\isacharasterisk}\ k\ mod\ {\isacharparenleft}n\ {\isacharasterisk}\ k{\isacharparenright}%
   124 \end{isabelle}
   125 \rulename{mod_mult_distrib}
   126 
   127 \begin{isabelle}%
   128 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}%
   129 \end{isabelle}
   130 \rulename{nat_diff_split}%
   131 \end{isamarkuptext}%
   132 \isamarkuptrue%
   133 \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
   134 \isamarkupfalse%
   135 \isacommand{apply}\ {\isacharparenleft}clarsimp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split\ iff\ del{\isacharcolon}\ less{\isacharunderscore}Suc{\isadigit{0}}{\isacharparenright}\isanewline
   136 \ %
   137 \isamarkupcmt{\begin{isabelle}%
   138 \ {\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}}%
   139 \end{isabelle}%
   140 }
   141 \isanewline
   142 \isamarkupfalse%
   143 \isacommand{apply}\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}n{\isacharequal}{\isadigit{0}}{\isachardoublequote}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline
   144 \isamarkupfalse%
   145 \isacommand{done}\isanewline
   146 \isanewline
   147 \isanewline
   148 \isamarkupfalse%
   149 \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
   150 \isamarkupfalse%
   151 \isacommand{apply}\ {\isacharparenleft}simp\ split{\isacharcolon}\ nat{\isacharunderscore}diff{\isacharunderscore}split{\isacharcomma}\ clarify{\isacharparenright}\isanewline
   152 \ %
   153 \isamarkupcmt{\begin{isabelle}%
   154 \ {\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}}%
   155 \end{isabelle}%
   156 }
   157 \isanewline
   158 \isamarkupfalse%
   159 \isacommand{apply}\ {\isacharparenleft}subgoal{\isacharunderscore}tac\ {\isachardoublequote}n{\isacharequal}{\isadigit{0}}\ {\isacharbar}\ n{\isacharequal}{\isadigit{1}}{\isachardoublequote}{\isacharcomma}\ force{\isacharcomma}\ arith{\isacharparenright}\isanewline
   160 \isamarkupfalse%
   161 \isacommand{done}\isamarkupfalse%
   162 %
   163 \begin{isamarkuptext}%
   164 \begin{isabelle}%
   165 m\ mod\ n\ {\isacharequal}\ {\isacharparenleft}if\ m\ {\isacharless}\ n\ then\ m\ else\ {\isacharparenleft}m\ {\isacharminus}\ n{\isacharparenright}\ mod\ n{\isacharparenright}%
   166 \end{isabelle}
   167 \rulename{mod_if}
   168 
   169 \begin{isabelle}%
   170 m\ div\ n\ {\isacharasterisk}\ n\ {\isacharplus}\ m\ mod\ n\ {\isacharequal}\ m%
   171 \end{isabelle}
   172 \rulename{mod_div_equality}
   173 
   174 
   175 \begin{isabelle}%
   176 a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c%
   177 \end{isabelle}
   178 \rulename{div_mult1_eq}
   179 
   180 \begin{isabelle}%
   181 a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c%
   182 \end{isabelle}
   183 \rulename{mod_mult1_eq}
   184 
   185 \begin{isabelle}%
   186 a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c%
   187 \end{isabelle}
   188 \rulename{div_mult2_eq}
   189 
   190 \begin{isabelle}%
   191 a\ mod\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ b\ {\isacharasterisk}\ {\isacharparenleft}a\ div\ b\ mod\ c{\isacharparenright}\ {\isacharplus}\ a\ mod\ b%
   192 \end{isabelle}
   193 \rulename{mod_mult2_eq}
   194 
   195 \begin{isabelle}%
   196 {\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ c\ {\isacharasterisk}\ a\ div\ {\isacharparenleft}c\ {\isacharasterisk}\ b{\isacharparenright}\ {\isacharequal}\ a\ div\ b%
   197 \end{isabelle}
   198 \rulename{div_mult_mult1}
   199 
   200 \begin{isabelle}%
   201 a\ div\ {\isadigit{0}}\ {\isacharequal}\ {\isadigit{0}}%
   202 \end{isabelle}
   203 \rulename{DIVISION_BY_ZERO_DIV}
   204 
   205 \begin{isabelle}%
   206 a\ mod\ {\isadigit{0}}\ {\isacharequal}\ a%
   207 \end{isabelle}
   208 \rulename{DIVISION_BY_ZERO_MOD}
   209 
   210 \begin{isabelle}%
   211 {\isasymlbrakk}m\ dvd\ n{\isacharsemicolon}\ n\ dvd\ m{\isasymrbrakk}\ {\isasymLongrightarrow}\ m\ {\isacharequal}\ n%
   212 \end{isabelle}
   213 \rulename{dvd_anti_sym}
   214 
   215 \begin{isabelle}%
   216 {\isasymlbrakk}k\ dvd\ m{\isacharsemicolon}\ k\ dvd\ n{\isasymrbrakk}\ {\isasymLongrightarrow}\ k\ dvd\ m\ {\isacharplus}\ n%
   217 \end{isabelle}
   218 \rulename{dvd_add}
   219 
   220 For the integers, I'd list a few theorems that somehow involve negative 
   221 numbers.%
   222 \end{isamarkuptext}%
   223 \isamarkuptrue%
   224 %
   225 \begin{isamarkuptext}%
   226 Division, remainder of negatives
   227 
   228 
   229 \begin{isabelle}%
   230 {\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isadigit{0}}\ {\isasymle}\ a\ mod\ b%
   231 \end{isabelle}
   232 \rulename{pos_mod_sign}
   233 
   234 \begin{isabelle}%
   235 {\isadigit{0}}\ {\isacharless}\ b\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isacharless}\ b%
   236 \end{isabelle}
   237 \rulename{pos_mod_bound}
   238 
   239 \begin{isabelle}%
   240 b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ a\ mod\ b\ {\isasymle}\ {\isadigit{0}}%
   241 \end{isabelle}
   242 \rulename{neg_mod_sign}
   243 
   244 \begin{isabelle}%
   245 b\ {\isacharless}\ {\isadigit{0}}\ {\isasymLongrightarrow}\ b\ {\isacharless}\ a\ mod\ b%
   246 \end{isabelle}
   247 \rulename{neg_mod_bound}
   248 
   249 \begin{isabelle}%
   250 {\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%
   251 \end{isabelle}
   252 \rulename{zdiv_zadd1_eq}
   253 
   254 \begin{isabelle}%
   255 {\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ mod\ c\ {\isacharequal}\ {\isacharparenleft}a\ mod\ c\ {\isacharplus}\ b\ mod\ c{\isacharparenright}\ mod\ c%
   256 \end{isabelle}
   257 \rulename{zmod_zadd1_eq}
   258 
   259 \begin{isabelle}%
   260 a\ {\isacharasterisk}\ b\ div\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ div\ c{\isacharparenright}\ {\isacharplus}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ div\ c%
   261 \end{isabelle}
   262 \rulename{zdiv_zmult1_eq}
   263 
   264 \begin{isabelle}%
   265 a\ {\isacharasterisk}\ b\ mod\ c\ {\isacharequal}\ a\ {\isacharasterisk}\ {\isacharparenleft}b\ mod\ c{\isacharparenright}\ mod\ c%
   266 \end{isabelle}
   267 \rulename{zmod_zmult1_eq}
   268 
   269 \begin{isabelle}%
   270 {\isadigit{0}}\ {\isacharless}\ c\ {\isasymLongrightarrow}\ a\ div\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}\ {\isacharequal}\ a\ div\ b\ div\ c%
   271 \end{isabelle}
   272 \rulename{zdiv_zmult2_eq}
   273 
   274 \begin{isabelle}%
   275 {\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%
   276 \end{isabelle}
   277 \rulename{zmod_zmult2_eq}
   278 
   279 \begin{isabelle}%
   280 {\isasymbar}a\ {\isacharasterisk}\ b{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharasterisk}\ {\isasymbar}b{\isasymbar}%
   281 \end{isabelle}
   282 \rulename{abs_mult}%
   283 \end{isamarkuptext}%
   284 \isamarkuptrue%
   285 \isacommand{lemma}\ {\isachardoublequote}abs\ {\isacharparenleft}x{\isacharplus}y{\isacharparenright}\ {\isasymle}\ abs\ x\ {\isacharplus}\ abs\ {\isacharparenleft}y\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline
   286 \isamarkupfalse%
   287 \isacommand{by}\ arith\isanewline
   288 \isanewline
   289 \isamarkupfalse%
   290 \isacommand{lemma}\ {\isachardoublequote}abs\ {\isacharparenleft}{\isadigit{2}}{\isacharasterisk}x{\isacharparenright}\ {\isacharequal}\ {\isadigit{2}}\ {\isacharasterisk}\ abs\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ int{\isacharparenright}{\isachardoublequote}\isanewline
   291 \isamarkupfalse%
   292 \isacommand{by}\ {\isacharparenleft}simp\ add{\isacharcolon}\ zabs{\isacharunderscore}def{\isacharparenright}\isamarkupfalse%
   293 %
   294 \begin{isamarkuptext}%
   295 Induction rules for the Integers
   296 
   297 \begin{isabelle}%
   298 {\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%
   299 \end{isabelle}
   300 \rulename{int_ge_induct}
   301 
   302 \begin{isabelle}%
   303 {\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%
   304 \end{isabelle}
   305 \rulename{int_gr_induct}
   306 
   307 \begin{isabelle}%
   308 {\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%
   309 \end{isabelle}
   310 \rulename{int_le_induct}
   311 
   312 \begin{isabelle}%
   313 {\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%
   314 \end{isabelle}
   315 \rulename{int_less_induct}%
   316 \end{isamarkuptext}%
   317 \isamarkuptrue%
   318 %
   319 \begin{isamarkuptext}%
   320 REALS
   321 
   322 \begin{isabelle}%
   323 a\ {\isacharless}\ b\ {\isasymLongrightarrow}\ {\isasymexists}r{\isachardot}\ a\ {\isacharless}\ r\ {\isasymand}\ r\ {\isacharless}\ b%
   324 \end{isabelle}
   325 \rulename{dense}
   326 
   327 \begin{isabelle}%
   328 {\isasymbar}a\ {\isacharcircum}\ n{\isasymbar}\ {\isacharequal}\ {\isasymbar}a{\isasymbar}\ {\isacharcircum}\ n%
   329 \end{isabelle}
   330 \rulename{power_abs}
   331 
   332 \begin{isabelle}%
   333 a\ {\isacharasterisk}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ b\ {\isacharslash}\ c%
   334 \end{isabelle}
   335 \rulename{times_divide_eq_right}
   336 
   337 \begin{isabelle}%
   338 b\ {\isacharslash}\ c\ {\isacharasterisk}\ a\ {\isacharequal}\ b\ {\isacharasterisk}\ a\ {\isacharslash}\ c%
   339 \end{isabelle}
   340 \rulename{times_divide_eq_left}
   341 
   342 \begin{isabelle}%
   343 a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharslash}\ c{\isacharparenright}\ {\isacharequal}\ a\ {\isacharasterisk}\ c\ {\isacharslash}\ b%
   344 \end{isabelle}
   345 \rulename{divide_divide_eq_right}
   346 
   347 \begin{isabelle}%
   348 a\ {\isacharslash}\ b\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharparenleft}b\ {\isacharasterisk}\ c{\isacharparenright}%
   349 \end{isabelle}
   350 \rulename{divide_divide_eq_left}
   351 
   352 \begin{isabelle}%
   353 {\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharminus}\ a\ {\isacharslash}\ b%
   354 \end{isabelle}
   355 \rulename{minus_divide_left}
   356 
   357 \begin{isabelle}%
   358 {\isacharminus}\ {\isacharparenleft}a\ {\isacharslash}\ b{\isacharparenright}\ {\isacharequal}\ a\ {\isacharslash}\ {\isacharminus}\ b%
   359 \end{isabelle}
   360 \rulename{minus_divide_right}
   361 
   362 This last NOT a simprule
   363 
   364 \begin{isabelle}%
   365 {\isacharparenleft}a\ {\isacharplus}\ b{\isacharparenright}\ {\isacharslash}\ c\ {\isacharequal}\ a\ {\isacharslash}\ c\ {\isacharplus}\ b\ {\isacharslash}\ c%
   366 \end{isabelle}
   367 \rulename{add_divide_distrib}%
   368 \end{isamarkuptext}%
   369 \isamarkuptrue%
   370 \isacommand{lemma}\ {\isachardoublequote}{\isadigit{3}}{\isacharslash}{\isadigit{4}}\ {\isacharless}\ {\isacharparenleft}{\isadigit{7}}{\isacharslash}{\isadigit{8}}\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}\isanewline
   371 \isamarkupfalse%
   372 \isacommand{by}\ simp\ \isanewline
   373 \isanewline
   374 \isamarkupfalse%
   375 \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%
   376 %
   377 \begin{isamarkuptxt}%
   378 \begin{isabelle}%
   379 \ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}{\isacharparenright}%
   380 \end{isabelle}%
   381 \end{isamarkuptxt}%
   382 \isamarkuptrue%
   383 \isacommand{apply}\ simp\isamarkupfalse%
   384 %
   385 \begin{isamarkuptxt}%
   386 \begin{isabelle}%
   387 \ {\isadigit{1}}{\isachardot}\ P\ {\isacharparenleft}{\isadigit{2}}\ {\isacharslash}\ {\isadigit{5}}{\isacharparenright}%
   388 \end{isabelle}%
   389 \end{isamarkuptxt}%
   390 \isamarkuptrue%
   391 \isacommand{oops}\isanewline
   392 \isanewline
   393 \isamarkupfalse%
   394 \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%
   395 %
   396 \begin{isamarkuptxt}%
   397 \begin{isabelle}%
   398 \ {\isadigit{1}}{\isachardot}\ {\isadigit{3}}\ {\isacharslash}\ {\isadigit{4}}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{8}}\ {\isacharslash}\ {\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ x%
   399 \end{isabelle}%
   400 \end{isamarkuptxt}%
   401 \isamarkuptrue%
   402 \isacommand{apply}\ simp\isamarkupfalse%
   403 %
   404 \begin{isamarkuptxt}%
   405 \begin{isabelle}%
   406 \ {\isadigit{1}}{\isachardot}\ {\isadigit{2}}\ {\isacharless}\ x\ {\isacharasterisk}\ {\isadigit{5}}%
   407 \end{isabelle}%
   408 \end{isamarkuptxt}%
   409 \isamarkuptrue%
   410 \isacommand{oops}\isanewline
   411 \isanewline
   412 \isamarkupfalse%
   413 \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isadigit{3}}{\isacharslash}{\isadigit{4}}{\isacharparenright}\ {\isacharasterisk}\ {\isacharparenleft}{\isadigit{1}}{\isadigit{0}}{\isacharcircum}{\isadigit{1}}{\isadigit{5}}{\isacharparenright}\ {\isacharless}\ {\isacharparenleft}x\ {\isacharcolon}{\isacharcolon}\ real{\isacharparenright}{\isachardoublequote}\isanewline
   414 \isamarkupfalse%
   415 \isacommand{apply}\ simp\ \isanewline
   416 \isamarkupfalse%
   417 \isacommand{oops}\isanewline
   418 \isanewline
   419 \isanewline
   420 \isanewline
   421 \isamarkupfalse%
   422 \isacommand{end}\isanewline
   423 \isamarkupfalse%
   424 \end{isabellebody}%
   425 %%% Local Variables:
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   427 %%% TeX-master: "root"
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